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

Percentage Accurate: 99.3% → 99.3%

Time: 54.8s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(\left(\cos x \cdot -1.5\right) \cdot \left(1 - \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + (((cos(x) * -1.5) * (1.0 - sqrt(5.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(y)))))) / (3.0d0 + (((cos(x) * (-1.5d0)) * (1.0d0 - sqrt(5.0d0))) + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.cos(x) - Math.cos(y)))))) / (3.0 + (((Math.cos(x) * -1.5) * (1.0 - Math.sqrt(5.0))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.cos(x) - math.cos(y)))))) / (3.0 + (((math.cos(x) * -1.5) * (1.0 - math.sqrt(5.0))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))))) / Float64(3.0 + Float64(Float64(Float64(cos(x) * -1.5) * Float64(1.0 - sqrt(5.0))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + (((cos(x) * -1.5) * (1.0 - sqrt(5.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * -1.5), $MachinePrecision] * N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * 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%

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

   1. associate-/l* 99.2%

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

   2. div-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. sub-neg 99.1%

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

   5. flip-- 99.0%

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

   6. metadata-eval 99.0%

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

   7. metadata-eval 99.0%

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

   8. associate-/r/ 99.1%

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

   9. metadata-eval 99.1%

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

   10. sub-neg 99.1%

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

   11. pow1/2 99.1%

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

   12. pow1/2 99.1%

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

   13. pow-sqr 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
6. Step-by-step derivation

   1. associate-/r* 99.4%

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

   2. metadata-eval 99.4%

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

   3. +-commutative 99.4%

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

7. Simplified 99.4%

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

   1. flip-- 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   4. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   5. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   6. pow-sqr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   9. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   10. metadata-eval 99.4%

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

   11. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. associate-*l/ 99.4%

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

    3. *-commutative 99.4%

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

    4. pow1 99.4%

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

    5. flip-+ 99.3%

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

    6. associate-/r/ 99.4%

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

    7. metadata-eval 99.4%

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

    8. sub-neg 99.4%

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

    9. pow1/2 99.4%

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

    10. pow1/2 99.4%

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

    11. pow-sqr 99.4%

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

    12. metadata-eval 99.4%

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

    13. metadata-eval 99.4%

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

    14. metadata-eval 99.4%

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

    15. metadata-eval 99.4%

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

    16. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

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

    1. unpow1 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(\left(\cos x \cdot -1.5\right) \cdot \left(1 - \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / (1.0 + sqrt(5.0))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.cos(x) - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.cos(x) - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0))))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / (1.0 + sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + 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%

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

   1. associate-/l* 99.2%

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

   2. div-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. sub-neg 99.1%

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

   5. flip-- 99.0%

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

   6. metadata-eval 99.0%

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

   7. metadata-eval 99.0%

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

   8. associate-/r/ 99.1%

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

   9. metadata-eval 99.1%

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

   10. sub-neg 99.1%

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

   11. pow1/2 99.1%

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

   12. pow1/2 99.1%

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

   13. pow-sqr 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
6. Step-by-step derivation

   1. associate-/r* 99.4%

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

   2. metadata-eval 99.4%

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

   3. +-commutative 99.4%

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

7. Simplified 99.4%

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

   1. flip-- 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   4. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   5. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   6. pow-sqr 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   9. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

   10. metadata-eval 99.4%

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

   11. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

11. Final simplification 99.4%

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(y)))))) / (3.0d0 - (1.5d0 * ((cos(x) * (1.0d0 - sqrt(5.0d0))) + (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around inf 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-out 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

   5. +-commutative 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around inf 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + -1\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \cos x \cdot \left(t_1 - 0.5\right)\\ \mathbf{if}\;x \leq -0.00315 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(t_2 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right) \cdot \left(\sin y \cdot t_0 + -0.0625 \cdot \left(x \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(t_2 - \cos y \cdot \left(t_1 - 1.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (cos y) -1.0))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (* (cos x) (- t_1 0.5))))
   (if (or (<= x -0.00315) (not (<= x 5.6e-11)))
     (/
      (-
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (- (cos y) (cos x)))))
      (* 3.0 (+ 1.0 (+ t_2 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0))))))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (/ (sin y) 16.0) (sin x)))
        (+ (* (sin y) t_0) (* -0.0625 (* x t_0)))))
      (* 3.0 (+ 1.0 (- t_2 (* (cos y) (- t_1 1.5)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) + -1.0;
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = cos(x) * (t_1 - 0.5);
	double tmp;
	if ((x <= -0.00315) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + (t_2 + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(y) / 16.0) - sin(x))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + (t_2 - (cos(y) * (t_1 - 1.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(y) + (-1.0d0)
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = cos(x) * (t_1 - 0.5d0)
    if ((x <= (-0.00315d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 - ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(y) - cos(x))))) / (3.0d0 * (1.0d0 + (t_2 + (cos(y) * (2.0d0 / (3.0d0 + sqrt(5.0d0)))))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((sin(y) / 16.0d0) - sin(x))) * ((sin(y) * t_0) + ((-0.0625d0) * (x * t_0))))) / (3.0d0 * (1.0d0 + (t_2 - (cos(y) * (t_1 - 1.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(y) + -1.0;
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = Math.cos(x) * (t_1 - 0.5);
	double tmp;
	if ((x <= -0.00315) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(y) - Math.cos(x))))) / (3.0 * (1.0 + (t_2 + (Math.cos(y) * (2.0 / (3.0 + Math.sqrt(5.0)))))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * ((Math.sin(y) / 16.0) - Math.sin(x))) * ((Math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + (t_2 - (Math.cos(y) * (t_1 - 1.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(y) + -1.0
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = math.cos(x) * (t_1 - 0.5)
	tmp = 0
	if (x <= -0.00315) or not (x <= 5.6e-11):
		tmp = (2.0 - ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(y) - math.cos(x))))) / (3.0 * (1.0 + (t_2 + (math.cos(y) * (2.0 / (3.0 + math.sqrt(5.0)))))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * ((math.sin(y) / 16.0) - math.sin(x))) * ((math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + (t_2 - (math.cos(y) * (t_1 - 1.5)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(y) + -1.0)
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(cos(x) * Float64(t_1 - 0.5))
	tmp = 0.0
	if ((x <= -0.00315) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 - Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(y) - cos(x))))) / Float64(3.0 * Float64(1.0 + Float64(t_2 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(y) / 16.0) - sin(x))) * Float64(Float64(sin(y) * t_0) + Float64(-0.0625 * Float64(x * t_0))))) / Float64(3.0 * Float64(1.0 + Float64(t_2 - Float64(cos(y) * Float64(t_1 - 1.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(y) + -1.0;
	t_1 = sqrt(5.0) / 2.0;
	t_2 = cos(x) * (t_1 - 0.5);
	tmp = 0.0;
	if ((x <= -0.00315) || ~((x <= 5.6e-11)))
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + (t_2 + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	else
		tmp = (2.0 + ((sqrt(2.0) * ((sin(y) / 16.0) - sin(x))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + (t_2 - (cos(y) * (t_1 - 1.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00315], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 - N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$2 - N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00315 or 5.6e-11 < x

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

      1. metadata-eval 99.0%

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

      2. div-sub 99.0%

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

      3. div-inv 99.0%

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

      4. flip-- 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l/ 98.9%

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

      7. sub-neg 98.9%

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

      8. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      9. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      10. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      11. pow-sqr 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      12. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{\color{blue}{1}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      13. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      14. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \color{blue}{-5}\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      15. metadata-eval 99.0%

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

      16. metadata-eval 99.0%

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

      17. +-commutative 99.0%

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

   8. Applied egg-rr 65.5%

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

   if -0.00315 < x < 5.6e-11

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00315 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right) + -0.0625 \cdot \left(x \cdot \left(\cos y + -1\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) - \cos y \cdot \left(\frac{\sqrt{5}}{2} - 1.5\right)\right)\right)}\\ \end{array} \]

Alternative 6: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) - \cos y \cdot \left(t_0 - 1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0054) (not (<= x 5.6e-11)))
     (/
      (-
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (- (cos y) (cos x)))))
      (* 3.0 (+ 1.0 (- (* (cos x) (- t_0 0.5)) (* (cos y) (- t_0 1.5))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (+ (sin x) (* -0.0625 (sin y)))
         (* (- 1.0 (cos y)) (+ (sin y) (* x -0.0625))))))
      (+
       3.0
       (+
        (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))
        (* 1.5 (* (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 <= -0.0054) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0054d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 - ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(y) - cos(x))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) - (cos(y) * (t_0 - 1.5d0)))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((1.0d0 - cos(y)) * (sin(y) + (x * (-0.0625d0))))))) / (3.0d0 + ((6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))) + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0054) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(y) - Math.cos(x))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) - (Math.cos(y) * (t_0 - 1.5)))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((1.0 - Math.cos(y)) * (Math.sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)))) + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0054) or not (x <= 5.6e-11):
		tmp = (2.0 - ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(y) - math.cos(x))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) - (math.cos(y) * (t_0 - 1.5)))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((1.0 - math.cos(y)) * (math.sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))) + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0054) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 - Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(y) - cos(x))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) - Float64(cos(y) * Float64(t_0 - 1.5))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(1.0 - cos(y)) * Float64(sin(y) + Float64(x * -0.0625)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0)))) + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0054) || ~((x <= 5.6e-11)))
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5)))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0054], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 - N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(x * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * 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 < -0.0054000000000000003 or 5.6e-11 < x

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   if -0.0054000000000000003 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.6%

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

   9. Taylor expanded in x around 0 99.3%

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

       1. +-commutative 99.3%

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

       2. associate-*r* 99.3%

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

       3. distribute-rgt-out 99.3%

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

   11. Simplified 99.3%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot x\right)\right)}\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0054 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) - \cos y \cdot \left(\frac{\sqrt{5}}{2} - 1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0024 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.0024) (not (<= x 5.6e-11)))
   (/
    (-
     2.0
     (*
      (* (sqrt 2.0) (sin x))
      (* (- (sin y) (/ (sin x) 16.0)) (- (cos y) (cos x)))))
    (*
     3.0
     (+
      1.0
      (+
       (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
       (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0))))))))
   (/
    (+
     2.0
     (*
      (sqrt 2.0)
      (*
       (+ (sin x) (* -0.0625 (sin y)))
       (* (- 1.0 (cos y)) (+ (sin y) (* x -0.0625))))))
    (+
     3.0
     (+
      (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))
      (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.0024) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.0024d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 - ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(y) - cos(x))))) / (3.0d0 * (1.0d0 + ((cos(x) * ((sqrt(5.0d0) / 2.0d0) - 0.5d0)) + (cos(y) * (2.0d0 / (3.0d0 + sqrt(5.0d0)))))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((1.0d0 - cos(y)) * (sin(y) + (x * (-0.0625d0))))))) / (3.0d0 + ((6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))) + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.0024) || !(x <= 5.6e-11)) {
		tmp = (2.0 - ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(y) - Math.cos(x))))) / (3.0 * (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) - 0.5)) + (Math.cos(y) * (2.0 / (3.0 + Math.sqrt(5.0)))))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((1.0 - Math.cos(y)) * (Math.sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)))) + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.0024) or not (x <= 5.6e-11):
		tmp = (2.0 - ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(y) - math.cos(x))))) / (3.0 * (1.0 + ((math.cos(x) * ((math.sqrt(5.0) / 2.0) - 0.5)) + (math.cos(y) * (2.0 / (3.0 + math.sqrt(5.0)))))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((1.0 - math.cos(y)) * (math.sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))) + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.0024) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 - Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(y) - cos(x))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(1.0 - cos(y)) * Float64(sin(y) + Float64(x * -0.0625)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0)))) + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.0024) || ~((x <= 5.6e-11)))
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.0024], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 - N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(x * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * 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 < -0.00239999999999999979 or 5.6e-11 < x

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

      1. metadata-eval 99.0%

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

      2. div-sub 99.0%

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

      3. div-inv 99.0%

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

      4. flip-- 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l/ 98.9%

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

      7. sub-neg 98.9%

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

      8. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      9. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      10. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      11. pow-sqr 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      12. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{\color{blue}{1}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      13. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      14. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \color{blue}{-5}\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      15. metadata-eval 99.0%

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

      16. metadata-eval 99.0%

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

      17. +-commutative 99.0%

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

   8. Applied egg-rr 65.5%

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

   if -0.00239999999999999979 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.6%

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

   9. Taylor expanded in x around 0 99.3%

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

       1. +-commutative 99.3%

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

       2. associate-*r* 99.3%

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

       3. distribute-rgt-out 99.3%

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

   11. Simplified 99.3%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot x\right)\right)}\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0024 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x + -0.0625 \cdot \sin y\\ t_1 := 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\\ t_2 := \cos y + -1\\ t_3 := 3 + \left(t_1 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\\ \mathbf{if}\;y \leq -0.052:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t_2\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_1\right)}\\ \mathbf{elif}\;y \leq 0.0012:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(t_0 \cdot \left(\sin y \cdot t_2\right)\right)}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sin x) (* -0.0625 (sin y))))
        (t_1 (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0)))))
        (t_2 (+ (cos y) -1.0))
        (t_3 (+ 3.0 (+ t_1 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))
   (if (<= y -0.052)
     (/
      (- 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_2))))
      (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_1)))
     (if (<= y 0.0012)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* t_0 (* (+ (cos x) -1.0) (+ y (* (sin x) -0.0625))))))
        t_3)
       (/ (- 2.0 (* (sqrt 2.0) (* t_0 (* (sin y) t_2)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = sin(x) + (-0.0625 * sin(y));
	double t_1 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	double t_2 = cos(y) + -1.0;
	double t_3 = 3.0 + (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	double tmp;
	if (y <= -0.052) {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_2)))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_1));
	} else if (y <= 0.0012) {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625)))))) / t_3;
	} else {
		tmp = (2.0 - (sqrt(2.0) * (t_0 * (sin(y) * t_2)))) / t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(x) + ((-0.0625d0) * sin(y))
    t_1 = 6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))
    t_2 = cos(y) + (-1.0d0)
    t_3 = 3.0d0 + (t_1 + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))
    if (y <= (-0.052d0)) then
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * t_2)))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + t_1))
    else if (y <= 0.0012d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_0 * ((cos(x) + (-1.0d0)) * (y + (sin(x) * (-0.0625d0))))))) / t_3
    else
        tmp = (2.0d0 - (sqrt(2.0d0) * (t_0 * (sin(y) * t_2)))) / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) + (-0.0625 * Math.sin(y));
	double t_1 = 6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)));
	double t_2 = Math.cos(y) + -1.0;
	double t_3 = 3.0 + (t_1 + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))));
	double tmp;
	if (y <= -0.052) {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * t_2)))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + t_1));
	} else if (y <= 0.0012) {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_0 * ((Math.cos(x) + -1.0) * (y + (Math.sin(x) * -0.0625)))))) / t_3;
	} else {
		tmp = (2.0 - (Math.sqrt(2.0) * (t_0 * (Math.sin(y) * t_2)))) / t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) + (-0.0625 * math.sin(y))
	t_1 = 6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))
	t_2 = math.cos(y) + -1.0
	t_3 = 3.0 + (t_1 + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))
	tmp = 0
	if y <= -0.052:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * t_2)))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + t_1))
	elif y <= 0.0012:
		tmp = (2.0 + (math.sqrt(2.0) * (t_0 * ((math.cos(x) + -1.0) * (y + (math.sin(x) * -0.0625)))))) / t_3
	else:
		tmp = (2.0 - (math.sqrt(2.0) * (t_0 * (math.sin(y) * t_2)))) / t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) + Float64(-0.0625 * sin(y)))
	t_1 = Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0))))
	t_2 = Float64(cos(y) + -1.0)
	t_3 = Float64(3.0 + Float64(t_1 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))
	tmp = 0.0
	if (y <= -0.052)
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_2)))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_1)));
	elseif (y <= 0.0012)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(Float64(cos(x) + -1.0) * Float64(y + Float64(sin(x) * -0.0625)))))) / t_3);
	else
		tmp = Float64(Float64(2.0 - Float64(sqrt(2.0) * Float64(t_0 * Float64(sin(y) * t_2)))) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) + (-0.0625 * sin(y));
	t_1 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	t_2 = cos(y) + -1.0;
	t_3 = 3.0 + (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	tmp = 0.0;
	if (y <= -0.052)
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * t_2)))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_1));
	elseif (y <= 0.0012)
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625)))))) / t_3;
	else
		tmp = (2.0 - (sqrt(2.0) * (t_0 * (sin(y) * t_2)))) / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 + N[(t$95$1 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.052], N[(N[(2.0 - N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0012], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(2.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0519999999999999976

   1. Initial program 99.2%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. div-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. sub-neg 98.8%

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

      5. flip-- 98.6%

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

      6. metadata-eval 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-/r/ 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. pow1/2 98.8%

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

      12. pow1/2 98.8%

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

      13. pow-sqr 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

      18. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.2%

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

      2. metadata-eval 99.2%

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

      3. +-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.3%

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

      11. +-commutative 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around inf 99.3%

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

   11. Taylor expanded in x around 0 61.0%

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

   if -0.0519999999999999976 < y < 0.00119999999999999989

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. associate-/l* 99.4%

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

      2. div-inv 99.4%

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

      3. metadata-eval 99.4%

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

      4. sub-neg 99.4%

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

      5. flip-- 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. associate-/r/ 99.4%

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

      9. metadata-eval 99.4%

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

      10. sub-neg 99.4%

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

      11. pow1/2 99.4%

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

      12. pow1/2 99.4%

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

      13. pow-sqr 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 99.5%

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

   9. Taylor expanded in y around 0 98.7%

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

       1. associate-*r* 98.7%

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

       2. *-commutative 98.7%

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

       3. sub-neg 98.7%

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

       4. metadata-eval 98.7%

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

       5. sub-neg 98.7%

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

       6. metadata-eval 98.7%

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

       7. distribute-rgt-out 98.7%

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

       8. *-commutative 98.7%

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

   11. Simplified 98.7%

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

   if 0.00119999999999999989 < y

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

      1. associate-/l* 99.0%

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

      2. div-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. sub-neg 98.8%

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

      5. flip-- 98.5%

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

      6. metadata-eval 98.5%

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

      7. metadata-eval 98.5%

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

      8. associate-/r/ 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. pow1/2 98.8%

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

      12. pow1/2 98.8%

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

      13. pow-sqr 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 99.2%

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

   9. Taylor expanded in x around 0 62.1%

      \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.052:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{elif}\;y \leq 0.0012:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 9: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0023 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) - \cos y \cdot \left(t_0 - 1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0023) (not (<= x 5.6e-11)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (+ (cos x) -1.0))))
      (* 3.0 (+ 1.0 (- (* (cos x) (- t_0 0.5)) (* (cos y) (- t_0 1.5))))))
     (/
      (-
       2.0
       (*
        (sqrt 2.0)
        (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) (+ (cos y) -1.0)))))
      (+
       3.0
       (+
        (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))
        (* 1.5 (* (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 <= -0.0023) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5)))));
	} else {
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0023d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(x) + (-1.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) - (cos(y) * (t_0 - 1.5d0)))))
    else
        tmp = (2.0d0 - (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))) + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0023) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) + -1.0)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) - (Math.cos(y) * (t_0 - 1.5)))));
	} else {
		tmp = (2.0 - (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) * (Math.cos(y) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)))) + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0023) or not (x <= 5.6e-11):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) + -1.0)))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) - (math.cos(y) * (t_0 - 1.5)))))
	else:
		tmp = (2.0 - (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) * (math.cos(y) + -1.0))))) / (3.0 + ((6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))) + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0023) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) - Float64(cos(y) * Float64(t_0 - 1.5))))));
	else
		tmp = Float64(Float64(2.0 - Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0)))) + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0023) || ~((x <= 5.6e-11)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5)))));
	else
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (1.0 + sqrt(5.0)))) + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0023], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * 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 < -0.0023 or 5.6e-11 < x

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in y around 0 62.1%

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

   if -0.0023 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.6%

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

   9. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) - \cos y \cdot \left(\frac{\sqrt{5}}{2} - 1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) - \cos y \cdot \left(t_0 - 1.5\right)\right)\right)\\ \mathbf{if}\;x \leq -0.002 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sin y \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1
         (* 3.0 (+ 1.0 (- (* (cos x) (- t_0 0.5)) (* (cos y) (- t_0 1.5)))))))
   (if (or (<= x -0.002) (not (<= x 5.6e-11)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (+ (cos x) -1.0))))
      t_1)
     (/
      (-
       2.0
       (*
        (* (sin y) (- 1.0 (cos y)))
        (* (sqrt 2.0) (- (/ (sin y) 16.0) (sin x)))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5))));
	double tmp;
	if ((x <= -0.002) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / t_1;
	} else {
		tmp = (2.0 - ((sin(y) * (1.0 - cos(y))) * (sqrt(2.0) * ((sin(y) / 16.0) - sin(x))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) - (cos(y) * (t_0 - 1.5d0))))
    if ((x <= (-0.002d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(x) + (-1.0d0))))) / t_1
    else
        tmp = (2.0d0 - ((sin(y) * (1.0d0 - cos(y))) * (sqrt(2.0d0) * ((sin(y) / 16.0d0) - sin(x))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) - (Math.cos(y) * (t_0 - 1.5))));
	double tmp;
	if ((x <= -0.002) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) + -1.0)))) / t_1;
	} else {
		tmp = (2.0 - ((Math.sin(y) * (1.0 - Math.cos(y))) * (Math.sqrt(2.0) * ((Math.sin(y) / 16.0) - Math.sin(x))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) - (math.cos(y) * (t_0 - 1.5))))
	tmp = 0
	if (x <= -0.002) or not (x <= 5.6e-11):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) + -1.0)))) / t_1
	else:
		tmp = (2.0 - ((math.sin(y) * (1.0 - math.cos(y))) * (math.sqrt(2.0) * ((math.sin(y) / 16.0) - math.sin(x))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) - Float64(cos(y) * Float64(t_0 - 1.5)))))
	tmp = 0.0
	if ((x <= -0.002) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) + -1.0)))) / t_1);
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(sin(y) * Float64(1.0 - cos(y))) * Float64(sqrt(2.0) * Float64(Float64(sin(y) / 16.0) - sin(x))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) - (cos(y) * (t_0 - 1.5))));
	tmp = 0.0;
	if ((x <= -0.002) || ~((x <= 5.6e-11)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / t_1;
	else
		tmp = (2.0 - ((sin(y) * (1.0 - cos(y))) * (sqrt(2.0) * ((sin(y) / 16.0) - sin(x))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 - 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.002], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-3 or 5.6e-11 < x

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in y around 0 62.1%

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

   if -2e-3 < x < 5.6e-11

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.002 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) - \cos y \cdot \left(\frac{\sqrt{5}}{2} - 1.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\sin y \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\frac{\sin y}{16} - \sin x\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) - \cos y \cdot \left(\frac{\sqrt{5}}{2} - 1.5\right)\right)\right)}\\ \end{array} \]

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(t_0 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))))
   (if (or (<= x -0.0011) (not (<= x 5.6e-11)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_0)))
     (/
      (-
       2.0
       (*
        (sqrt 2.0)
        (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) (+ (cos y) -1.0)))))
      (+ 3.0 (+ t_0 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	double tmp;
	if ((x <= -0.0011) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_0));
	} else {
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + (t_0 + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))
    if ((x <= (-0.0011d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + t_0))
    else
        tmp = (2.0d0 - (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) * (cos(y) + (-1.0d0)))))) / (3.0d0 + (t_0 + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)));
	double tmp;
	if ((x <= -0.0011) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + t_0));
	} else {
		tmp = (2.0 - (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) * (Math.cos(y) + -1.0))))) / (3.0 + (t_0 + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))
	tmp = 0
	if (x <= -0.0011) or not (x <= 5.6e-11):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + t_0))
	else:
		tmp = (2.0 - (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) * (math.cos(y) + -1.0))))) / (3.0 + (t_0 + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0))))
	tmp = 0.0
	if ((x <= -0.0011) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_0)));
	else
		tmp = Float64(Float64(2.0 - Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(t_0 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	tmp = 0.0;
	if ((x <= -0.0011) || ~((x <= 5.6e-11)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_0));
	else
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + (t_0 + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0011], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(t$95$0 + N[(1.5 * 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 < -0.00110000000000000007 or 5.6e-11 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. div-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. sub-neg 98.8%

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

      5. flip-- 98.5%

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

      6. metadata-eval 98.5%

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

      7. metadata-eval 98.5%

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

      8. associate-/r/ 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. pow1/2 98.8%

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

      12. pow1/2 98.8%

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

      13. pow-sqr 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. metadata-eval 99.1%

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

      18. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in y around 0 61.6%

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

   if -0.00110000000000000007 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.6%

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

   9. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.0024:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \cos x \cdot \left(0.5 - t_0\right)\right) - t_0\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(t_1 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))))
   (if (<= x -0.0024)
     (/
      (-
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (- (cos y) (cos x)))))
      (* 3.0 (+ 1.0 (- (- 1.5 (* (cos x) (- 0.5 t_0))) t_0))))
     (if (<= x 5.6e-11)
       (/
        (-
         2.0
         (*
          (sqrt 2.0)
          (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) (+ (cos y) -1.0)))))
        (+ 3.0 (+ t_1 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0)))))))
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	double tmp;
	if (x <= -0.0024) {
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((1.5 - (cos(x) * (0.5 - t_0))) - t_0)));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = 6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))
    if (x <= (-0.0024d0)) then
        tmp = (2.0d0 - ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(y) - cos(x))))) / (3.0d0 * (1.0d0 + ((1.5d0 - (cos(x) * (0.5d0 - t_0))) - t_0)))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 - (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) * (cos(y) + (-1.0d0)))))) / (3.0d0 + (t_1 + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0))))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = 6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)));
	double tmp;
	if (x <= -0.0024) {
		tmp = (2.0 - ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(y) - Math.cos(x))))) / (3.0 * (1.0 + ((1.5 - (Math.cos(x) * (0.5 - t_0))) - t_0)));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) * (Math.cos(y) + -1.0))))) / (3.0 + (t_1 + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = 6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))
	tmp = 0
	if x <= -0.0024:
		tmp = (2.0 - ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(y) - math.cos(x))))) / (3.0 * (1.0 + ((1.5 - (math.cos(x) * (0.5 - t_0))) - t_0)))
	elif x <= 5.6e-11:
		tmp = (2.0 - (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) * (math.cos(y) + -1.0))))) / (3.0 + (t_1 + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0))))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0))))
	tmp = 0.0
	if (x <= -0.0024)
		tmp = Float64(Float64(2.0 - Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(y) - cos(x))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(1.5 - Float64(cos(x) * Float64(0.5 - t_0))) - t_0))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 - Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(t_1 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = 6.0 * (cos(x) / (1.0 + sqrt(5.0)));
	tmp = 0.0;
	if (x <= -0.0024)
		tmp = (2.0 - ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(y) - cos(x))))) / (3.0 * (1.0 + ((1.5 - (cos(x) * (0.5 - t_0))) - t_0)));
	elseif (x <= 5.6e-11)
		tmp = (2.0 - (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (cos(y) + -1.0))))) / (3.0 + (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0024], N[(N[(2.0 - N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(1.5 - N[(N[Cos[x], $MachinePrecision] * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 - N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(t$95$1 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00239999999999999979

   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. associate-+l+ 98.8%

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

      3. *-commutative 98.8%

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

      4. div-sub 98.8%

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

      5. metadata-eval 98.8%

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

      6. *-commutative 98.8%

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

      7. div-sub 98.8%

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

      8. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 62.5%

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

      1. *-commutative 62.5%

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

   6. Simplified 62.5%

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

   7. Taylor expanded in y around 0 58.8%

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

   if -0.00239999999999999979 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.6%

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

   9. Taylor expanded in x around 0 99.2%

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

   if 5.6e-11 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

      1. associate-/l* 99.0%

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

      2. div-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. sub-neg 98.9%

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

      5. flip-- 98.6%

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

      6. metadata-eval 98.6%

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

      7. metadata-eval 98.6%

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

      8. associate-/r/ 98.9%

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

      9. metadata-eval 98.9%

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

      10. sub-neg 98.9%

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

      11. pow1/2 98.9%

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

      12. pow1/2 98.9%

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

      13. pow-sqr 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. metadata-eval 99.3%

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

      3. +-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. flip-- 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in y around 0 64.1%

       \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0024:\\ \;\;\;\;\frac{2 - \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos y - \cos x\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 - \cos x \cdot \left(0.5 - \sqrt{5} \cdot 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 13: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \sqrt{5}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{t_0} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (sqrt 5.0))))
   (if (or (<= x -1.5e-6) (not (<= x 5.6e-11)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+
       3.0
       (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 6.0 (/ (cos x) t_0)))))
     (/
      (- 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
      (+ 3.0 (- (/ 6.0 t_0) (* 1.5 (* (cos y) (- (sqrt 5.0) 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + sqrt(5.0);
	double tmp;
	if ((x <= -1.5e-6) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / t_0))));
	} else {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / t_0) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + sqrt(5.0d0)
    if ((x <= (-1.5d-6)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (6.0d0 * (cos(x) / t_0))))
    else
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 / t_0) - (1.5d0 * (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.sqrt(5.0);
	double tmp;
	if ((x <= -1.5e-6) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (6.0 * (Math.cos(x) / t_0))));
	} else {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / (3.0 + ((6.0 / t_0) - (1.5 * (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.sqrt(5.0)
	tmp = 0
	if (x <= -1.5e-6) or not (x <= 5.6e-11):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (6.0 * (math.cos(x) / t_0))))
	else:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / (3.0 + ((6.0 / t_0) - (1.5 * (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + sqrt(5.0))
	tmp = 0.0
	if ((x <= -1.5e-6) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(6.0 * Float64(cos(x) / t_0)))));
	else
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 / t_0) - Float64(1.5 * Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + sqrt(5.0);
	tmp = 0.0;
	if ((x <= -1.5e-6) || ~((x <= 5.6e-11)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / t_0))));
	else
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / t_0) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.5e-6], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[x], $MachinePrecision] / 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[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 / t$95$0), $MachinePrecision] - N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-6 or 5.6e-11 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. div-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. sub-neg 98.8%

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

      5. flip-- 98.5%

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

      6. metadata-eval 98.5%

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

      7. metadata-eval 98.5%

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

      8. associate-/r/ 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. pow1/2 98.8%

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

      12. pow1/2 98.8%

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

      13. pow-sqr 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. metadata-eval 99.1%

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

      18. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in y around 0 61.6%

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

   if -1.5e-6 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. flip-- 99.3%

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

      2. metadata-eval 99.3%

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

      3. metadata-eval 99.3%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. pow1/2 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-sqr 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{6}{1 + \sqrt{5}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \end{array} \]

Alternative 14: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)\\ \mathbf{if}\;x \leq -0.00092 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (+
           (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
           (* 6.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))))))
   (if (or (<= x -0.00092) (not (<= x 5.6e-11)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      t_0)
     (/
      (- 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / (1.0 + sqrt(5.0)))));
	double tmp;
	if ((x <= -0.00092) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / t_0;
	} else {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (6.0d0 * (cos(x) / (1.0d0 + sqrt(5.0d0)))))
    if ((x <= (-0.00092d0)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / t_0
    else
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (6.0 * (Math.cos(x) / (1.0 + Math.sqrt(5.0)))));
	double tmp;
	if ((x <= -0.00092) || !(x <= 5.6e-11)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / t_0;
	} else {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (6.0 * (math.cos(x) / (1.0 + math.sqrt(5.0)))))
	tmp = 0
	if (x <= -0.00092) or not (x <= 5.6e-11):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / t_0
	else:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(6.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0))))))
	tmp = 0.0
	if ((x <= -0.00092) || !(x <= 5.6e-11))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / t_0);
	else
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (6.0 * (cos(x) / (1.0 + sqrt(5.0)))));
	tmp = 0.0;
	if ((x <= -0.00092) || ~((x <= 5.6e-11)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / t_0;
	else
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00092], N[Not[LessEqual[x, 5.6e-11]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$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[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000003e-4 or 5.6e-11 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/l* 98.9%

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

      2. div-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. sub-neg 98.8%

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

      5. flip-- 98.5%

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

      6. metadata-eval 98.5%

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

      7. metadata-eval 98.5%

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

      8. associate-/r/ 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. pow1/2 98.8%

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

      12. pow1/2 98.8%

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

      13. pow-sqr 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. metadata-eval 99.1%

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

      18. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in y around 0 61.6%

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

   if -9.2000000000000003e-4 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. flip-- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r/ 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

      11. pow1/2 99.5%

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

      12. pow1/2 99.5%

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

      13. pow-sqr 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. flip-- 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.6%

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

      11. +-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in x around 0 99.0%

       \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \left(6 \cdot \frac{\cos x}{1 + \sqrt{5}} + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00092 \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 15: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ t_2 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_0 \cdot t_1\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2 \cdot \frac{1}{t_2}\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot t_0\right) \cdot t_1}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \frac{4}{t_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (+ (cos x) -1.0)))
        (t_2 (+ 3.0 (sqrt 5.0))))
   (if (<= x -4.8e-6)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* t_0 t_1)))
       (+ 1.0 (+ (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)) (* 2.0 (/ 1.0 t_2))))))
     (if (<= x 5.6e-11)
       (/
        (-
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
        (+
         3.0
         (-
          (/ 6.0 (+ 1.0 (sqrt 5.0)))
          (* 1.5 (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/
        (+ 2.0 (* (* -0.0625 t_0) t_1))
        (+ 3.0 (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (/ 4.0 t_2)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * (cos(x) + -1.0);
	double t_2 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -4.8e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * t_1))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (2.0 * (1.0 / t_2)))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + ((-0.0625 * t_0) * t_1)) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 / t_2))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = sqrt(2.0d0) * (cos(x) + (-1.0d0))
    t_2 = 3.0d0 + sqrt(5.0d0)
    if (x <= (-4.8d-6)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_0 * t_1))) / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)) + (2.0d0 * (1.0d0 / t_2)))))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 / (1.0d0 + sqrt(5.0d0))) - (1.5d0 * (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = (2.0d0 + (((-0.0625d0) * t_0) * t_1)) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (4.0d0 / t_2))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.sqrt(2.0) * (Math.cos(x) + -1.0);
	double t_2 = 3.0 + Math.sqrt(5.0);
	double tmp;
	if (x <= -4.8e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * t_1))) / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)) + (2.0 * (1.0 / t_2)))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + Math.sqrt(5.0))) - (1.5 * (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + ((-0.0625 * t_0) * t_1)) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (4.0 / t_2))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.sqrt(2.0) * (math.cos(x) + -1.0)
	t_2 = 3.0 + math.sqrt(5.0)
	tmp = 0
	if x <= -4.8e-6:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * t_1))) / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)) + (2.0 * (1.0 / t_2)))))
	elif x <= 5.6e-11:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + math.sqrt(5.0))) - (1.5 * (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = (2.0 + ((-0.0625 * t_0) * t_1)) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (4.0 / t_2))))
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	t_2 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -4.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * t_1))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(2.0 * Float64(1.0 / t_2))))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 / Float64(1.0 + sqrt(5.0))) - Float64(1.5 * Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * t_0) * t_1)) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(4.0 / t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = sqrt(2.0) * (cos(x) + -1.0);
	t_2 = 3.0 + sqrt(5.0);
	tmp = 0.0;
	if (x <= -4.8e-6)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * t_1))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (2.0 * (1.0 / t_2)))));
	elseif (x <= 5.6e-11)
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = (2.0 + ((-0.0625 * t_0) * t_1)) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 / t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 - N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999998e-6

   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. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. associate-*l* 98.8%

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

      4. fma-def 98.9%

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

      5. +-commutative 98.9%

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

      6. associate-+r+ 98.8%

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

   3. Simplified 99.0%

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

      1. metadata-eval 99.0%

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

      2. div-sub 99.0%

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

      3. div-inv 99.0%

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

      4. flip-- 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-*l/ 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      9. pow1/2 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      10. pow1/2 98.8%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      11. pow-sqr 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      12. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-{5}^{\color{blue}{1}}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      13. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \left(-\color{blue}{5}\right)\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      14. metadata-eval 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \mathsf{fma}\left(\cos y, \frac{\left(9 + \color{blue}{-5}\right) \cdot 0.5}{3 + \sqrt{5}}, 1\right)\right)} \]

      15. metadata-eval 99.0%

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

      16. metadata-eval 99.0%

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

      17. +-commutative 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 57.6%

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

   if -4.7999999999999998e-6 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. flip-- 99.3%

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

      2. metadata-eval 99.3%

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

      3. metadata-eval 99.3%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. pow1/2 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-sqr 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 5.6e-11 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 62.9%

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

      1. associate-*r* 62.9%

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

      2. *-commutative 62.9%

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

      3. sub-neg 62.9%

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

      4. metadata-eval 62.9%

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

      5. distribute-lft-out 62.9%

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

      6. sub-neg 62.9%

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

      7. metadata-eval 62.9%

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

   6. Simplified 62.9%

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

      1. flip-- 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   8. Applied egg-rr 63.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 16: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.25e-6) (not (<= x 5.6e-11)))
   (/
    (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
    (+
     3.0
     (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))
   (/
    (- 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
    (+
     3.0
     (- (/ 6.0 (+ 1.0 (sqrt 5.0))) (* 1.5 (* (cos y) (- (sqrt 5.0) 3.0))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.25e-6) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.25d-6)) .or. (.not. (x <= 5.6d-11))) then
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (4.0d0 / (3.0d0 + sqrt(5.0d0))))))
    else
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 / (1.0d0 + sqrt(5.0d0))) - (1.5d0 * (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.25e-6) || !(x <= 5.6e-11)) {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (4.0 / (3.0 + Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + Math.sqrt(5.0))) - (1.5 * (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.25e-6) or not (x <= 5.6e-11):
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (4.0 / (3.0 + math.sqrt(5.0))))))
	else:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + math.sqrt(5.0))) - (1.5 * (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.25e-6) || !(x <= 5.6e-11))
		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(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(4.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 / Float64(1.0 + sqrt(5.0))) - Float64(1.5 * Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.25e-6) || ~((x <= 5.6e-11)))
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 / (3.0 + sqrt(5.0))))));
	else
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.25e-6], N[Not[LessEqual[x, 5.6e-11]], $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[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2499999999999998e-6 or 5.6e-11 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 60.4%

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

      1. associate-*r* 60.4%

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

      2. *-commutative 60.4%

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

      3. sub-neg 60.4%

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

      4. metadata-eval 60.4%

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

      5. distribute-lft-out 60.4%

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

      6. sub-neg 60.4%

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

      7. metadata-eval 60.4%

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

   6. Simplified 60.4%

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

      1. flip-- 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      5. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      6. pow-sqr 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{{5}^{\left(2 \cdot 0.5\right)}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-{5}^{\color{blue}{1}}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      9. metadata-eval 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \cos x \cdot \frac{6}{1 + \sqrt{5}}\right)} \]

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   8. Applied egg-rr 60.5%

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

   if -3.2499999999999998e-6 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. flip-- 99.3%

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

      2. metadata-eval 99.3%

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

      3. metadata-eval 99.3%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. pow1/2 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-sqr 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{6}{1 + \sqrt{5}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-6} \lor \neg \left(x \leq 5.6 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \end{array} \]

Alternative 17: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_2 := {\sin x}^{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot t_2\right) \cdot t_0}{1 + 0.5 \cdot \left(t_1 + t_3\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot t_0\right)}{3 + \left(1.5 \cdot t_1 + 1.5 \cdot t_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (+ (cos x) -1.0)))
        (t_1 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -5.6e-5)
     (*
      0.3333333333333333
      (/ (+ 2.0 (* (* -0.0625 t_2) t_0)) (+ 1.0 (* 0.5 (+ t_1 t_3)))))
     (if (<= x 5.6e-11)
       (/
        (+
         2.0
         (* (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y)))))
        (- 3.0 (* 1.5 (+ (- 1.0 (sqrt 5.0)) (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/
        (+ 2.0 (* -0.0625 (* t_2 t_0)))
        (+ 3.0 (+ (* 1.5 t_1) (* 1.5 t_3))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (cos(x) + -1.0);
	double t_1 = cos(x) * (sqrt(5.0) + -1.0);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -5.6e-5) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * pow(sin(y), 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (cos(x) + (-1.0d0))
    t_1 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_2 = sin(x) ** 2.0d0
    t_3 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-5.6d-5)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (((-0.0625d0) * t_2) * t_0)) / (1.0d0 + (0.5d0 * (t_1 + t_3))))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 + (((-0.0625d0) * (sin(y) ** 2.0d0)) * (sqrt(2.0d0) * (1.0d0 - cos(y))))) / (3.0d0 - (1.5d0 * ((1.0d0 - sqrt(5.0d0)) + (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (t_2 * t_0))) / (3.0d0 + ((1.5d0 * t_1) + (1.5d0 * t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * (Math.cos(x) + -1.0);
	double t_1 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -5.6e-5) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(y), 2.0)) * (Math.sqrt(2.0) * (1.0 - Math.cos(y))))) / (3.0 - (1.5 * ((1.0 - Math.sqrt(5.0)) + (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * (math.cos(x) + -1.0)
	t_1 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -5.6e-5:
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))))
	elif x <= 5.6e-11:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(y), 2.0)) * (math.sqrt(2.0) * (1.0 - math.cos(y))))) / (3.0 - (1.5 * ((1.0 - math.sqrt(5.0)) + (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -5.6e-5)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(-0.0625 * t_2) * t_0)) / Float64(1.0 + Float64(0.5 * Float64(t_1 + t_3)))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))) / Float64(3.0 - Float64(1.5 * Float64(Float64(1.0 - sqrt(5.0)) + Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_0))) / Float64(3.0 + Float64(Float64(1.5 * t_1) + Float64(1.5 * t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * (cos(x) + -1.0);
	t_1 = cos(x) * (sqrt(5.0) + -1.0);
	t_2 = sin(x) ^ 2.0;
	t_3 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -5.6e-5)
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	elseif (x <= 5.6e-11)
		tmp = (2.0 + ((-0.0625 * (sin(y) ^ 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e-5], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(1.5 * N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * t$95$1), $MachinePrecision] + N[(1.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.59999999999999992e-5

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

      3. sub-neg 57.5%

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

      4. metadata-eval 57.5%

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

      5. distribute-lft-out 57.5%

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

      6. sub-neg 57.5%

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

      7. metadata-eval 57.5%

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

   4. Simplified 57.5%

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

   if -5.59999999999999992e-5 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-lft-out 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

   if 5.6e-11 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 62.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 18: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ t_1 := \sqrt{5} + -1\\ t_2 := \cos x \cdot t_1\\ t_3 := {\sin x}^{2}\\ t_4 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.3 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot t_3\right) \cdot t_0}{1 + 0.5 \cdot \left(t_2 + t_4\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(1.5 \cdot t_1 - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_3 \cdot t_0\right)}{3 + \left(1.5 \cdot t_2 + 1.5 \cdot t_4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (+ (cos x) -1.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (* (cos x) t_1))
        (t_3 (pow (sin x) 2.0))
        (t_4 (- 3.0 (sqrt 5.0))))
   (if (<= x -5.3e-6)
     (*
      0.3333333333333333
      (/ (+ 2.0 (* (* -0.0625 t_3) t_0)) (+ 1.0 (* 0.5 (+ t_2 t_4)))))
     (if (<= x 5.6e-11)
       (/
        (-
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
        (+ 3.0 (- (* 1.5 t_1) (* 1.5 (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/
        (+ 2.0 (* -0.0625 (* t_3 t_0)))
        (+ 3.0 (+ (* 1.5 t_2) (* 1.5 t_4))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (cos(x) + -1.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = cos(x) * t_1;
	double t_3 = pow(sin(x), 2.0);
	double t_4 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -5.3e-6) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_3) * t_0)) / (1.0 + (0.5 * (t_2 + t_4))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((1.5 * t_1) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_3 * t_0))) / (3.0 + ((1.5 * t_2) + (1.5 * t_4)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (cos(x) + (-1.0d0))
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = cos(x) * t_1
    t_3 = sin(x) ** 2.0d0
    t_4 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-5.3d-6)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (((-0.0625d0) * t_3) * t_0)) / (1.0d0 + (0.5d0 * (t_2 + t_4))))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((1.5d0 * t_1) - (1.5d0 * (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (t_3 * t_0))) / (3.0d0 + ((1.5d0 * t_2) + (1.5d0 * t_4)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * (Math.cos(x) + -1.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = Math.cos(x) * t_1;
	double t_3 = Math.pow(Math.sin(x), 2.0);
	double t_4 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -5.3e-6) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_3) * t_0)) / (1.0 + (0.5 * (t_2 + t_4))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / (3.0 + ((1.5 * t_1) - (1.5 * (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_3 * t_0))) / (3.0 + ((1.5 * t_2) + (1.5 * t_4)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * (math.cos(x) + -1.0)
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = math.cos(x) * t_1
	t_3 = math.pow(math.sin(x), 2.0)
	t_4 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -5.3e-6:
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_3) * t_0)) / (1.0 + (0.5 * (t_2 + t_4))))
	elif x <= 5.6e-11:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / (3.0 + ((1.5 * t_1) - (1.5 * (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = (2.0 + (-0.0625 * (t_3 * t_0))) / (3.0 + ((1.5 * t_2) + (1.5 * t_4)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(cos(x) * t_1)
	t_3 = sin(x) ^ 2.0
	t_4 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -5.3e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(-0.0625 * t_3) * t_0)) / Float64(1.0 + Float64(0.5 * Float64(t_2 + t_4)))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(1.5 * t_1) - Float64(1.5 * Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_3 * t_0))) / Float64(3.0 + Float64(Float64(1.5 * t_2) + Float64(1.5 * t_4))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * (cos(x) + -1.0);
	t_1 = sqrt(5.0) + -1.0;
	t_2 = cos(x) * t_1;
	t_3 = sin(x) ^ 2.0;
	t_4 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -5.3e-6)
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_3) * t_0)) / (1.0 + (0.5 * (t_2 + t_4))));
	elseif (x <= 5.6e-11)
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((1.5 * t_1) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = (2.0 + (-0.0625 * (t_3 * t_0))) / (3.0 + ((1.5 * t_2) + (1.5 * t_4)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.3e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(-0.0625 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 - N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * t$95$1), $MachinePrecision] - N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * t$95$2), $MachinePrecision] + N[(1.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3000000000000001e-6

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

      3. sub-neg 57.5%

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

      4. metadata-eval 57.5%

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

      5. distribute-lft-out 57.5%

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

      6. sub-neg 57.5%

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

      7. metadata-eval 57.5%

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

   4. Simplified 57.5%

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

   if -5.3000000000000001e-6 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

   if 5.6e-11 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 62.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\sqrt{5} + -1\right) - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 19: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_2 := {\sin x}^{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot t_2\right) \cdot t_0}{1 + 0.5 \cdot \left(t_1 + t_3\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot t_0\right)}{3 + \left(1.5 \cdot t_1 + 1.5 \cdot t_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (+ (cos x) -1.0)))
        (t_1 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -4e-7)
     (*
      0.3333333333333333
      (/ (+ 2.0 (* (* -0.0625 t_2) t_0)) (+ 1.0 (* 0.5 (+ t_1 t_3)))))
     (if (<= x 5.6e-11)
       (/
        (-
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (+ (cos y) -1.0)))))
        (+
         3.0
         (-
          (/ 6.0 (+ 1.0 (sqrt 5.0)))
          (* 1.5 (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/
        (+ 2.0 (* -0.0625 (* t_2 t_0)))
        (+ 3.0 (+ (* 1.5 t_1) (* 1.5 t_3))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (cos(x) + -1.0);
	double t_1 = cos(x) * (sqrt(5.0) + -1.0);
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4e-7) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (cos(x) + (-1.0d0))
    t_1 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_2 = sin(x) ** 2.0d0
    t_3 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-4d-7)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (((-0.0625d0) * t_2) * t_0)) / (1.0d0 + (0.5d0 * (t_1 + t_3))))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 - ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 / (1.0d0 + sqrt(5.0d0))) - (1.5d0 * (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (t_2 * t_0))) / (3.0d0 + ((1.5d0 * t_1) + (1.5d0 * t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * (Math.cos(x) + -1.0);
	double t_1 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -4e-7) {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 - (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + Math.sqrt(5.0))) - (1.5 * (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * (math.cos(x) + -1.0)
	t_1 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -4e-7:
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))))
	elif x <= 5.6e-11:
		tmp = (2.0 - (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + math.sqrt(5.0))) - (1.5 * (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(-0.0625 * t_2) * t_0)) / Float64(1.0 + Float64(0.5 * Float64(t_1 + t_3)))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 - Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 / Float64(1.0 + sqrt(5.0))) - Float64(1.5 * Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_0))) / Float64(3.0 + Float64(Float64(1.5 * t_1) + Float64(1.5 * t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * (cos(x) + -1.0);
	t_1 = cos(x) * (sqrt(5.0) + -1.0);
	t_2 = sin(x) ^ 2.0;
	t_3 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -4e-7)
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * t_2) * t_0)) / (1.0 + (0.5 * (t_1 + t_3))));
	elseif (x <= 5.6e-11)
		tmp = (2.0 - (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) + -1.0))))) / (3.0 + ((6.0 / (1.0 + sqrt(5.0))) - (1.5 * (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = (2.0 + (-0.0625 * (t_2 * t_0))) / (3.0 + ((1.5 * t_1) + (1.5 * t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e-7], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 - N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * t$95$1), $MachinePrecision] + N[(1.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999998e-7

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

      3. sub-neg 57.5%

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

      4. metadata-eval 57.5%

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

      5. distribute-lft-out 57.5%

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

      6. sub-neg 57.5%

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

      7. metadata-eval 57.5%

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

   4. Simplified 57.5%

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

   if -3.9999999999999998e-7 < x < 5.6e-11

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. flip-- 99.3%

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

      2. metadata-eval 99.3%

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

      3. metadata-eval 99.3%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. pow1/2 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-sqr 99.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \left(\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)\right)}{\sqrt{5} + 1}\right)} \]

      10. metadata-eval 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \left({5}^{\color{blue}{1}} + \left(-1\right)\right)}{\sqrt{5} + 1}\right)} \]

      11. metadata-eval 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \left(\color{blue}{5} + \left(-1\right)\right)}{\sqrt{5} + 1}\right)} \]

      12. metadata-eval 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \left(5 + \color{blue}{-1}\right)}{\sqrt{5} + 1}\right)} \]

      13. metadata-eval 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1.5 \cdot \color{blue}{4}}{\sqrt{5} + 1}\right)} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{\color{blue}{6}}{\sqrt{5} + 1}\right)} \]

      15. +-commutative 99.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{6}{\color{blue}{1 + \sqrt{5}}}\right)} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{6}{1 + \sqrt{5}}}\right)} \]

   if 5.6e-11 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in y around 0 62.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 - -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y + -1\right)\right)\right)}{3 + \left(\frac{6}{1 + \sqrt{5}} - 1.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 20: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(\left(3 + t_1\right) - \sqrt{5}\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot \left(t_1 + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (t_1 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (<= x -2.85e-6)
     (/ t_0 (+ 3.0 (* 1.5 (- (+ 3.0 t_1) (sqrt 5.0)))))
     (if (<= x 5.6e-11)
       (/
        (+
         2.0
         (* (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y)))))
        (- 3.0 (* 1.5 (+ (- 1.0 (sqrt 5.0)) (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/ t_0 (+ 3.0 (* 1.5 (+ t_1 (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)));
	double t_1 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -2.85e-6) {
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_1) - sqrt(5.0))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * pow(sin(y), 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * (t_1 + (3.0 - sqrt(5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))
    t_1 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    if (x <= (-2.85d-6)) then
        tmp = t_0 / (3.0d0 + (1.5d0 * ((3.0d0 + t_1) - sqrt(5.0d0))))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 + (((-0.0625d0) * (sin(y) ** 2.0d0)) * (sqrt(2.0d0) * (1.0d0 - cos(y))))) / (3.0d0 - (1.5d0 * ((1.0d0 - sqrt(5.0d0)) + (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = t_0 / (3.0d0 + (1.5d0 * (t_1 + (3.0d0 - sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)));
	double t_1 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -2.85e-6) {
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_1) - Math.sqrt(5.0))));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(y), 2.0)) * (Math.sqrt(2.0) * (1.0 - Math.cos(y))))) / (3.0 - (1.5 * ((1.0 - Math.sqrt(5.0)) + (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * (t_1 + (3.0 - Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))
	t_1 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	tmp = 0
	if x <= -2.85e-6:
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_1) - math.sqrt(5.0))))
	elif x <= 5.6e-11:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(y), 2.0)) * (math.sqrt(2.0) * (1.0 - math.cos(y))))) / (3.0 - (1.5 * ((1.0 - math.sqrt(5.0)) + (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = t_0 / (3.0 + (1.5 * (t_1 + (3.0 - math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if (x <= -2.85e-6)
		tmp = Float64(t_0 / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + t_1) - sqrt(5.0)))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))) / Float64(3.0 - Float64(1.5 * Float64(Float64(1.0 - sqrt(5.0)) + Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(t_0 / Float64(3.0 + Float64(1.5 * Float64(t_1 + Float64(3.0 - sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)));
	t_1 = cos(x) * (sqrt(5.0) + -1.0);
	tmp = 0.0;
	if (x <= -2.85e-6)
		tmp = t_0 / (3.0 + (1.5 * ((3.0 + t_1) - sqrt(5.0))));
	elseif (x <= 5.6e-11)
		tmp = (2.0 + ((-0.0625 * (sin(y) ^ 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = t_0 / (3.0 + (1.5 * (t_1 + (3.0 - sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.85e-6], N[(t$95$0 / N[(3.0 + N[(1.5 * N[(N[(3.0 + t$95$1), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(1.5 * N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(1.5 * N[(t$95$1 + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8499999999999998e-6

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in y around 0 57.3%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 57.3%

         \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sub-neg 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. metadata-eval 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. distribute-lft-out 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      6. sub-neg 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval 57.3%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

   6. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in x around inf 57.4%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{\left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}} \]

   if -2.8499999999999998e-6 < x < 5.6e-11

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 99.5%

         \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. distribute-lft-out 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}} \]

      4. sub-neg 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)} \]

      5. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + \color{blue}{-1}\right)\right)} \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)}} \]

   if 5.6e-11 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in y around 0 62.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 62.9%

         \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sub-neg 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. metadata-eval 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. distribute-lft-out 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      6. sub-neg 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 21: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + 0.5 \cdot t_1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + 1.5 \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (t_1 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (- 3.0 (sqrt 5.0)))))
   (if (<= x -1.8e-6)
     (* 0.3333333333333333 (/ t_0 (+ 1.0 (* 0.5 t_1))))
     (if (<= x 5.6e-11)
       (/
        (+
         2.0
         (* (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y)))))
        (- 3.0 (* 1.5 (+ (- 1.0 (sqrt 5.0)) (* (cos y) (- (sqrt 5.0) 3.0))))))
       (/ t_0 (+ 3.0 (* 1.5 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)));
	double t_1 = (cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0));
	double tmp;
	if (x <= -1.8e-6) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * t_1)));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * pow(sin(y), 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))
    t_1 = (cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (3.0d0 - sqrt(5.0d0))
    if (x <= (-1.8d-6)) then
        tmp = 0.3333333333333333d0 * (t_0 / (1.0d0 + (0.5d0 * t_1)))
    else if (x <= 5.6d-11) then
        tmp = (2.0d0 + (((-0.0625d0) * (sin(y) ** 2.0d0)) * (sqrt(2.0d0) * (1.0d0 - cos(y))))) / (3.0d0 - (1.5d0 * ((1.0d0 - sqrt(5.0d0)) + (cos(y) * (sqrt(5.0d0) - 3.0d0)))))
    else
        tmp = t_0 / (3.0d0 + (1.5d0 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)));
	double t_1 = (Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (3.0 - Math.sqrt(5.0));
	double tmp;
	if (x <= -1.8e-6) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * t_1)));
	} else if (x <= 5.6e-11) {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(y), 2.0)) * (Math.sqrt(2.0) * (1.0 - Math.cos(y))))) / (3.0 - (1.5 * ((1.0 - Math.sqrt(5.0)) + (Math.cos(y) * (Math.sqrt(5.0) - 3.0)))));
	} else {
		tmp = t_0 / (3.0 + (1.5 * t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))
	t_1 = (math.cos(x) * (math.sqrt(5.0) + -1.0)) + (3.0 - math.sqrt(5.0))
	tmp = 0
	if x <= -1.8e-6:
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * t_1)))
	elif x <= 5.6e-11:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(y), 2.0)) * (math.sqrt(2.0) * (1.0 - math.cos(y))))) / (3.0 - (1.5 * ((1.0 - math.sqrt(5.0)) + (math.cos(y) * (math.sqrt(5.0) - 3.0)))))
	else:
		tmp = t_0 / (3.0 + (1.5 * t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))
	t_1 = Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(3.0 - sqrt(5.0)))
	tmp = 0.0
	if (x <= -1.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + Float64(0.5 * t_1))));
	elseif (x <= 5.6e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))) / Float64(3.0 - Float64(1.5 * Float64(Float64(1.0 - sqrt(5.0)) + Float64(cos(y) * Float64(sqrt(5.0) - 3.0))))));
	else
		tmp = Float64(t_0 / Float64(3.0 + Float64(1.5 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)));
	t_1 = (cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0));
	tmp = 0.0;
	if (x <= -1.8e-6)
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * t_1)));
	elseif (x <= 5.6e-11)
		tmp = (2.0 + ((-0.0625 * (sin(y) ^ 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (3.0 - (1.5 * ((1.0 - sqrt(5.0)) + (cos(y) * (sqrt(5.0) - 3.0)))));
	else
		tmp = t_0 / (3.0 + (1.5 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e-6], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e-11], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[(1.5 * N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999992e-6

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0 57.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. associate-*r* 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sub-neg 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. metadata-eval 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. distribute-lft-out 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{1 + \color{blue}{0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      6. sub-neg 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval 57.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

   4. Simplified 57.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   if -1.79999999999999992e-6 < x < 5.6e-11

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 99.5%

         \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. *-commutative 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. distribute-lft-out 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}} \]

      4. sub-neg 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)} \]

      5. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + \color{blue}{-1}\right)\right)} \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)}} \]

   if 5.6e-11 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in y around 0 62.9%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 62.9%

         \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. sub-neg 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. metadata-eval 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. distribute-lft-out 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      6. sub-neg 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

      7. metadata-eval 62.9%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 - 1.5 \cdot \left(\left(1 - \sqrt{5}\right) + \cos y \cdot \left(\sqrt{5} - 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 22: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
  (+ 3.0 (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (- 3.0 (sqrt 5.0)))))))

C

double code(double x, double y) {
	return (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (3.0d0 - sqrt(5.0d0)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (3.0 - Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (3.0 - math.sqrt(5.0)))))

Julia

function code(x, y)
	return 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(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(3.0 - sqrt(5.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0)))));
end

Wolfram

code[x_, y_] := 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[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in y around 0 62.7%

   \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 62.7%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. *-commutative 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. distribute-lft-out 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   6. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

6. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

7. Final simplification 62.7%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)} \]

Alternative 23: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
  (+ 3.0 (* 1.5 (+ 3.0 (- (* (cos x) (+ (sqrt 5.0) -1.0)) (sqrt 5.0)))))))

C

double code(double x, double y) {
	return (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * (3.0 + ((cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 + (1.5d0 * (3.0d0 + ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) - sqrt(5.0d0)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / (3.0 + (1.5 * (3.0 + ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) - Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / (3.0 + (1.5 * (3.0 + ((math.cos(x) * (math.sqrt(5.0) + -1.0)) - math.sqrt(5.0)))))

Julia

function code(x, y)
	return 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(1.5 * Float64(3.0 + Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) - sqrt(5.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 + (1.5 * (3.0 + ((cos(x) * (sqrt(5.0) + -1.0)) - sqrt(5.0)))));
end

Wolfram

code[x_, y_] := 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[(1.5 * N[(3.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in y around 0 62.7%

   \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 62.7%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. *-commutative 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. distribute-lft-out 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   6. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

6. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}} \]

   2. associate-+l- 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{\left(3 - \left(\sqrt{5} - \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}} \]

8. Applied egg-rr 62.7%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{\left(3 - \left(\sqrt{5} - \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}} \]

9. Final simplification 62.7%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(3 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) - \sqrt{5}\right)\right)} \]

Alternative 24: 40.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
  6.0))

C

double code(double x, double y) {
	return (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / 6.0;
end

Wolfram

code[x_, y_] := 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] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in y around 0 62.7%

   \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 62.7%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. *-commutative 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. distribute-lft-out 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   6. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

6. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

7. Taylor expanded in x around 0 44.2%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{2}} \]

8. Final simplification 44.2%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{6} \]

Alternative 25: 40.3% accurate, 1139.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in y around 0 62.7%

   \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 62.7%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. *-commutative 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. distribute-lft-out 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   6. sub-neg 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 62.7%

      \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

6. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

7. Taylor expanded in x around 0 44.2%

   \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)}{3 + 1.5 \cdot \color{blue}{2}} \]

8. Taylor expanded in x around 0 36.6%

   \[\leadsto \frac{2 + \color{blue}{0.03125 \cdot \left({x}^{4} \cdot \sqrt{2}\right)}}{3 + 1.5 \cdot 2} \]
9. Step-by-step derivation

   1. associate-*r* 36.6%

      \[\leadsto \frac{2 + \color{blue}{\left(0.03125 \cdot {x}^{4}\right) \cdot \sqrt{2}}}{3 + 1.5 \cdot 2} \]

   2. *-commutative 36.6%

      \[\leadsto \frac{2 + \color{blue}{\sqrt{2} \cdot \left(0.03125 \cdot {x}^{4}\right)}}{3 + 1.5 \cdot 2} \]

10. Simplified 36.6%

    \[\leadsto \frac{2 + \color{blue}{\sqrt{2} \cdot \left(0.03125 \cdot {x}^{4}\right)}}{3 + 1.5 \cdot 2} \]

11. Taylor expanded in x around 0 44.2%

    \[\leadsto \color{blue}{0.3333333333333333} \]

12. Final simplification 44.2%

    \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023301 
(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))))))