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

Percentage Accurate: 99.3% → 99.3%

Time: 48.1s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot 1.5\right) \cdot \left(\sqrt{5} + -1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* (sin x) -0.0625))
    (* (- (cos x) (cos y)) (+ (sin x) (* (sin y) -0.0625))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (* (/ 4.0 (+ 3.0 (sqrt 5.0))) 1.5)
    (* (* (cos x) 1.5) (+ (sqrt 5.0) -1.0))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * -0.0625)))), 2.0) / (3.0 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) * 1.5), ((cos(x) * 1.5) * (sqrt(5.0) + -1.0))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) + Float64(sin(y) * -0.0625)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) * 1.5), Float64(Float64(cos(x) * 1.5) * Float64(sqrt(5.0) + -1.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * 1.5), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 99.4%

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

    1. associate-*r* 99.4%

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

    2. sub-neg 99.4%

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

    3. metadata-eval 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 99.4%

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

    1. associate-*r* 99.4%

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

    2. sub-neg 99.4%

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

    3. metadata-eval 99.4%

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

11. Simplified 99.4%

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

12. Taylor expanded in y around inf 99.4%

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

    1. fma-define 99.4%

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

    2. *-commutative 99.4%

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

    3. sub-neg 99.4%

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

    4. metadata-eval 99.4%

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

    5. associate-*r/ 99.4%

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

    6. +-commutative 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]
16. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in y around inf 99.4%

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

10. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]
11. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (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) + (sin(y) * (-0.0625d0))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(y)))))) / (3.0d0 + ((1.5d0 * (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) + (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) + (Math.sin(y) * -0.0625)) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.cos(x) - Math.cos(y)))))) / (3.0 + ((1.5 * (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) + (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) + (math.sin(y) * -0.0625)) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.cos(x) - math.cos(y)))))) / (3.0 + ((1.5 * (math.cos(x) * (math.sqrt(5.0) + -1.0))) + (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(sin(y) * -0.0625)) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + 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) + (sin(y) * -0.0625)) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (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[(N[Sin[y], $MachinePrecision] * -0.0625), $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $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%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in y around inf 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.0043 \lor \neg \left(x \leq 0.0038\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_0\right)}{3 \cdot \left(t\_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(t\_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (or (<= x -0.0043) (not (<= x 0.0038)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) (sin x)) t_0)))
      (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (* (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))) (- 1.0 (cos y))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((x <= -0.0043) || !(x <= 0.0038)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    if ((x <= (-0.0043d0)) .or. (.not. (x <= 0.0038d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * t_0))) / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((t_0 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))) * (1.0d0 - cos(y)))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((x <= -0.0043) || !(x <= 0.0038)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((t_0 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (1.0 - Math.cos(y)))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	tmp = 0
	if (x <= -0.0043) or not (x <= 0.0038):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((t_0 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))) * (1.0 - math.cos(y)))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if ((x <= -0.0043) || !(x <= 0.0038))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	tmp = 0.0;
	if ((x <= -0.0043) || ~((x <= 0.0038)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0043], N[Not[LessEqual[x, 0.0038]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0043 or 0.00379999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.7%

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

      1. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   if -0.0043 < x < 0.00379999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0043 \lor \neg \left(x \leq 0.0038\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.00122 \lor \neg \left(x \leq 0.0056\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_0\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.00122) (not (<= x 0.0056)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) (sin x)) t_0))) t_1)
     (/
      (+
       2.0
       (* (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))) (- 1.0 (cos y))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00122) || !(x <= 0.0056)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.00122d0)) .or. (.not. (x <= 0.0056d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * t_0))) / t_1
    else
        tmp = (2.0d0 + ((t_0 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))) * (1.0d0 - cos(y)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00122) || !(x <= 0.0056)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (1.0 - Math.cos(y)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.00122) or not (x <= 0.0056):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / t_1
	else:
		tmp = (2.0 + ((t_0 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))) * (1.0 - math.cos(y)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.00122) || !(x <= 0.0056))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(1.0 - cos(y)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.00122) || ~((x <= 0.0056)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	else
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00122], N[Not[LessEqual[x, 0.0056]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00121999999999999995 or 0.00559999999999999994 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.7%

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

      1. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   if -0.00121999999999999995 < x < 0.00559999999999999994

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00122 \lor \neg \left(x \leq 0.0056\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0045 \lor \neg \left(x \leq 0.00255\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.0045) (not (<= x 0.00255)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      t_0)
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (+ (* -0.0625 (pow (sin y) 2.0)) (* x (* (sin y) 1.00390625))))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0045) || !(x <= 0.00255)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / t_0;
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * pow(sin(y), 2.0)) + (x * (sin(y) * 1.00390625)))))) / 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 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.0045d0)) .or. (.not. (x <= 0.00255d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / t_0
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (((-0.0625d0) * (sin(y) ** 2.0d0)) + (x * (sin(y) * 1.00390625d0)))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0045) || !(x <= 0.00255)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / t_0;
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * ((-0.0625 * Math.pow(Math.sin(y), 2.0)) + (x * (Math.sin(y) * 1.00390625)))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.0045) or not (x <= 0.00255):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / t_0
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * ((-0.0625 * math.pow(math.sin(y), 2.0)) + (x * (math.sin(y) * 1.00390625)))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.0045) || !(x <= 0.00255))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(x * Float64(sin(y) * 1.00390625)))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.0045) || ~((x <= 0.00255)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / t_0;
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * (sin(y) ^ 2.0)) + (x * (sin(y) * 1.00390625)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0045], N[Not[LessEqual[x, 0.00255]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00449999999999999966 or 0.0025500000000000002 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.7%

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

      1. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   if -0.00449999999999999966 < x < 0.0025500000000000002

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. distribute-rgt1-in 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0045 \lor \neg \left(x \leq 0.00255\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x + -1\\ t_2 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0037:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot t\_1}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t\_0 - 0.5\right)\right) - t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.0027:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_2 - 0.5\right) + \cos y \cdot \left(1.5 - t\_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (+ (cos x) -1.0))
        (t_2 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.0037)
     (/
      (+
       2.0
       (*
        (*
         (sqrt 2.0)
         (* (- (sin y) (/ (sin x) 16.0)) (- (sin x) (/ (sin y) 16.0))))
        t_1))
      (* 3.0 (+ 1.0 (- (+ 1.5 (* (cos x) (- t_0 0.5))) t_0))))
     (if (<= x 0.0027)
       (/
        (+
         2.0
         (*
          (- 1.0 (cos y))
          (*
           (sqrt 2.0)
           (+ (* -0.0625 (pow (sin y) 2.0)) (* x (* (sin y) 1.00390625))))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
       (/
        (+ 2.0 (* t_1 (* (* (sqrt 2.0) -0.0625) (pow (sin x) 2.0))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(x) + -1.0;
	double t_2 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.0037) {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0)))) * t_1)) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	} else if (x <= 0.0027) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * pow(sin(y), 2.0)) + (x * (sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * -0.0625) * pow(sin(x), 2.0)))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = cos(x) + (-1.0d0)
    t_2 = sqrt(5.0d0) / 2.0d0
    if (x <= (-0.0037d0)) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((sin(y) - (sin(x) / 16.0d0)) * (sin(x) - (sin(y) / 16.0d0)))) * t_1)) / (3.0d0 * (1.0d0 + ((1.5d0 + (cos(x) * (t_0 - 0.5d0))) - t_0)))
    else if (x <= 0.0027d0) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (((-0.0625d0) * (sin(y) ** 2.0d0)) + (x * (sin(y) * 1.00390625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + (t_1 * ((sqrt(2.0d0) * (-0.0625d0)) * (sin(x) ** 2.0d0)))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + (cos(y) * (1.5d0 - t_2)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = Math.cos(x) + -1.0;
	double t_2 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.0037) {
		tmp = (2.0 + ((Math.sqrt(2.0) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * t_1)) / (3.0 * (1.0 + ((1.5 + (Math.cos(x) * (t_0 - 0.5))) - t_0)));
	} else if (x <= 0.0027) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * ((-0.0625 * Math.pow(Math.sin(y), 2.0)) + (x * (Math.sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (t_1 * ((Math.sqrt(2.0) * -0.0625) * Math.pow(Math.sin(x), 2.0)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.cos(x) + -1.0
	t_2 = math.sqrt(5.0) / 2.0
	tmp = 0
	if x <= -0.0037:
		tmp = (2.0 + ((math.sqrt(2.0) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sin(x) - (math.sin(y) / 16.0)))) * t_1)) / (3.0 * (1.0 + ((1.5 + (math.cos(x) * (t_0 - 0.5))) - t_0)))
	elif x <= 0.0027:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * ((-0.0625 * math.pow(math.sin(y), 2.0)) + (x * (math.sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + (t_1 * ((math.sqrt(2.0) * -0.0625) * math.pow(math.sin(x), 2.0)))) / (3.0 * (1.0 + ((math.cos(x) * (t_2 - 0.5)) + (math.cos(y) * (1.5 - t_2)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(x) + -1.0)
	t_2 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.0037)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * t_1)) / Float64(3.0 * Float64(1.0 + Float64(Float64(1.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0))));
	elseif (x <= 0.0027)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(x) ^ 2.0)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = cos(x) + -1.0;
	t_2 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (x <= -0.0037)
		tmp = (2.0 + ((sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0)))) * t_1)) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	elseif (x <= 0.0027)
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * (sin(y) ^ 2.0)) + (x * (sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * -0.0625) * (sin(x) ^ 2.0)))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.0037], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0027], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0037000000000000002

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

      5. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 61.0%

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

   6. Taylor expanded in y around 0 60.0%

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

   if -0.0037000000000000002 < x < 0.0027000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. distribute-rgt1-in 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   if 0.0027000000000000001 < x

   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%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.8%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around 0 68.0%

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

   6. Taylor expanded in y around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*r* 67.0%

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

   8. Simplified 67.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0037:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.0027:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.0044 \lor \neg \left(y \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(y, \sin x \cdot 1.00390625, -0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t\_0 - 0.5\right)\right) - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (/ (sqrt 5.0) 2.0)))
   (if (or (<= y -0.0044) (not (<= y 9.5e-5)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin y) 2.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))
     (/
      (+
       2.0
       (*
        (+ (cos x) -1.0)
        (*
         (sqrt 2.0)
         (fma y (* (sin x) 1.00390625) (* -0.0625 (pow (sin x) 2.0))))))
      (* 3.0 (+ 1.0 (- (+ 1.5 (* (cos x) (- t_0 0.5))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -0.0044) || !(y <= 9.5e-5)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	} else {
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * fma(y, (sin(x) * 1.00390625), (-0.0625 * pow(sin(x), 2.0)))))) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.0044) || !(y <= 9.5e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * fma(y, Float64(sin(x) * 1.00390625), Float64(-0.0625 * (sin(x) ^ 2.0)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(1.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0044], N[Not[LessEqual[y, 9.5e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(y * N[(N[Sin[x], $MachinePrecision] * 1.00390625), $MachinePrecision] + N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00440000000000000027 or 9.5000000000000005e-5 < y

   1. Initial program 99.0%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 56.7%

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

   if -0.00440000000000000027 < y < 9.5000000000000005e-5

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

      2. distribute-rgt-in 99.5%

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

      3. cos-neg 99.5%

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

      4. distribute-rgt-in 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.8%

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

   6. Taylor expanded in y around 0 98.4%

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

   7. Taylor expanded in y around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. fma-define 98.4%

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

      3. distribute-rgt1-in 98.4%

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

      4. *-commutative 98.4%

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

      5. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0044 \lor \neg \left(y \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(y, \sin x \cdot 1.00390625, -0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot t\_2}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t\_0 - 0.5\right)\right) - t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (+ (cos x) -1.0)))
   (if (<= x -1.6e-6)
     (/
      (+
       2.0
       (*
        (*
         (sqrt 2.0)
         (* (- (sin y) (/ (sin x) 16.0)) (- (sin x) (/ (sin y) 16.0))))
        t_2))
      (* 3.0 (+ 1.0 (- (+ 1.5 (* (cos x) (- t_0 0.5))) t_0))))
     (if (<= x 1.1e-6)
       (/
        (fma -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+ (/ (* (cos y) 6.0) (+ 3.0 (sqrt 5.0))) (+ 1.5 (* (sqrt 5.0) 1.5))))
       (/
        (+ 2.0 (* t_2 (* (* (sqrt 2.0) -0.0625) (pow (sin x) 2.0))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = cos(x) + -1.0;
	double tmp;
	if (x <= -1.6e-6) {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0)))) * t_2)) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	} else if (x <= 1.1e-6) {
		tmp = fma(-0.0625, (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (((cos(y) * 6.0) / (3.0 + sqrt(5.0))) + (1.5 + (sqrt(5.0) * 1.5)));
	} else {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * -0.0625) * pow(sin(x), 2.0)))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -1.6e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * t_2)) / Float64(3.0 * Float64(1.0 + Float64(Float64(1.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0))));
	elseif (x <= 1.1e-6)
		tmp = Float64(fma(-0.0625, Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(Float64(Float64(cos(y) * 6.0) / Float64(3.0 + sqrt(5.0))) + Float64(1.5 + Float64(sqrt(5.0) * 1.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(x) ^ 2.0)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -1.6e-6], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-6], N[(N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5999999999999999e-6

   1. Initial program 99.0%

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

      1. associate-*l* 99.1%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.1%

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

      5. associate-+l+ 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 61.6%

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

   6. Taylor expanded in y around 0 60.6%

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

   if -1.5999999999999999e-6 < x < 1.1000000000000001e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. associate-*r* 99.7%

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

       2. sub-neg 99.7%

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

       3. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 99.7%

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

       1. fma-define 99.7%

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

       2. *-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

       5. associate-*r/ 99.7%

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

       6. +-commutative 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in x around 0 99.1%

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

       1. +-commutative 99.1%

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

       2. fma-define 99.1%

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

       3. associate-*r* 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-*l* 99.1%

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

       6. associate-+r+ 99.2%

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

       7. +-commutative 99.2%

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

       8. associate-*r/ 99.2%

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

       9. +-commutative 99.2%

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

       10. sub-neg 99.2%

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

       11. metadata-eval 99.2%

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

       12. +-commutative 99.2%

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

       13. distribute-rgt-in 99.2%

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

       14. metadata-eval 99.2%

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

       15. metadata-eval 99.2%

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

   17. Simplified 99.2%

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

   if 1.1000000000000001e-6 < x

   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%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.8%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around 0 68.0%

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

   6. Taylor expanded in y around 0 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*r* 67.0%

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

   8. Simplified 67.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-5} \lor \neg \left(y \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e-5) (not (<= y 8.5e-6)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))))
     (+
      1.0
      (+
       (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))
       (* 0.5 (* (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
     (fma 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)) (/ 6.0 (+ 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e-5) || !(y <= 8.5e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (pow(sin(y), 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (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 + fma(1.5, (cos(x) * (sqrt(5.0) + -1.0)), (6.0 / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e-5) || !(y <= 8.5e-6))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(0.5 * 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 + fma(1.5, Float64(cos(x) * Float64(sqrt(5.0) + -1.0)), Float64(6.0 / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e-5], N[Not[LessEqual[y, 8.5e-6]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000008e-5 or 8.4999999999999999e-6 < 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 99.0%

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

   5. Taylor expanded in y around inf 98.9%

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

   6. Taylor expanded in x around 0 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-*l* 56.3%

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

      3. associate-*r* 56.3%

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

      4. *-commutative 56.3%

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

      5. *-commutative 56.3%

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

      6. *-commutative 56.3%

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

      7. associate-*l* 56.3%

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

   8. Simplified 56.3%

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

   if -3.00000000000000008e-5 < y < 8.4999999999999999e-6

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 99.2%

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

       1. sub-neg 99.2%

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

       2. metadata-eval 99.2%

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

       3. fma-define 99.3%

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

       4. sub-neg 99.3%

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

       5. metadata-eval 99.3%

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

       6. associate-*r/ 99.3%

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

       7. metadata-eval 99.3%

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

       8. +-commutative 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-5} \lor \neg \left(y \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-6} \lor \neg \left(y \leq 2.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot t\_0, \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -1.85e-6) (not (<= y 2.05e-5)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin y) 2.0)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (fma 1.5 (* (cos x) t_0) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.85e-6) || !(y <= 2.05e-5)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + fma(1.5, (cos(x) * t_0), (6.0 / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -1.85e-6) || !(y <= 2.05e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	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 + fma(1.5, Float64(cos(x) * t_0), Float64(6.0 / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.85e-6], N[Not[LessEqual[y, 2.05e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8500000000000001e-6 or 2.05000000000000002e-5 < y

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0 57.5%

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

   4. Taylor expanded in x around 0 56.4%

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

   if -1.8500000000000001e-6 < y < 2.05000000000000002e-5

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 99.2%

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

       1. sub-neg 99.2%

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

       2. metadata-eval 99.2%

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

       3. fma-define 99.3%

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

       4. sub-neg 99.3%

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

       5. metadata-eval 99.3%

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

       6. associate-*r/ 99.3%

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

       7. metadata-eval 99.3%

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

       8. +-commutative 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-6} \lor \neg \left(y \leq 2.05 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -3.6e-6) (not (<= x 8.5e-6)))
     (/
      (+ 2.0 (* (+ (cos x) -1.0) (* (* (sqrt 2.0) -0.0625) (pow (sin x) 2.0))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (fma -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+ (/ (* (cos y) 6.0) (+ 3.0 (sqrt 5.0))) (+ 1.5 (* (sqrt 5.0) 1.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -3.6e-6) || !(x <= 8.5e-6)) {
		tmp = (2.0 + ((cos(x) + -1.0) * ((sqrt(2.0) * -0.0625) * pow(sin(x), 2.0)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = fma(-0.0625, (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (((cos(y) * 6.0) / (3.0 + sqrt(5.0))) + (1.5 + (sqrt(5.0) * 1.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -3.6e-6) || !(x <= 8.5e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(x) ^ 2.0)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(fma(-0.0625, Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(Float64(Float64(cos(y) * 6.0) / Float64(3.0 + sqrt(5.0))) + Float64(1.5 + Float64(sqrt(5.0) * 1.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -3.6e-6], N[Not[LessEqual[x, 8.5e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $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[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999984e-6 or 8.4999999999999999e-6 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 65.1%

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

   6. Taylor expanded in y around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*r* 63.7%

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

   8. Simplified 63.7%

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

   if -3.59999999999999984e-6 < x < 8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. associate-*r* 99.7%

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

       2. sub-neg 99.7%

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

       3. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 99.7%

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

       1. fma-define 99.7%

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

       2. *-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

       5. associate-*r/ 99.7%

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

       6. +-commutative 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in x around 0 99.1%

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

       1. +-commutative 99.1%

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

       2. fma-define 99.1%

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

       3. associate-*r* 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-*l* 99.1%

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

       6. associate-+r+ 99.2%

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

       7. +-commutative 99.2%

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

       8. associate-*r/ 99.2%

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

       9. +-commutative 99.2%

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

       10. sub-neg 99.2%

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

       11. metadata-eval 99.2%

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

       12. +-commutative 99.2%

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

       13. distribute-rgt-in 99.2%

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

       14. metadata-eval 99.2%

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

       15. metadata-eval 99.2%

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

   17. Simplified 99.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_1}{3 + \left(1.5 \cdot t\_2 + 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{t\_0} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + \mathsf{fma}\left(1.5, t\_2, \frac{6}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_2 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (<= x -1.6e-5)
     (/ t_1 (+ 3.0 (+ (* 1.5 t_2) (* 6.0 (/ 1.0 t_0)))))
     (if (<= x 7e-6)
       (/
        (fma -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+ (/ (* (cos y) 6.0) t_0) (+ 1.5 (* (sqrt 5.0) 1.5))))
       (/ t_1 (+ 3.0 (fma 1.5 t_2 (/ 6.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_2 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -1.6e-5) {
		tmp = t_1 / (3.0 + ((1.5 * t_2) + (6.0 * (1.0 / t_0))));
	} else if (x <= 7e-6) {
		tmp = fma(-0.0625, (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (((cos(y) * 6.0) / t_0) + (1.5 + (sqrt(5.0) * 1.5)));
	} else {
		tmp = t_1 / (3.0 + fma(1.5, t_2, (6.0 / t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_2 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if (x <= -1.6e-5)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(1.5 * t_2) + Float64(6.0 * Float64(1.0 / t_0)))));
	elseif (x <= 7e-6)
		tmp = Float64(fma(-0.0625, Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(Float64(Float64(cos(y) * 6.0) / t_0) + Float64(1.5 + Float64(sqrt(5.0) * 1.5))));
	else
		tmp = Float64(t_1 / Float64(3.0 + fma(1.5, t_2, Float64(6.0 / t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-5], N[(t$95$1 / N[(3.0 + N[(N[(1.5 * t$95$2), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-6], N[(N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * t$95$2 + N[(6.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999993e-5

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in y around 0 59.1%

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

   if -1.59999999999999993e-5 < x < 6.99999999999999989e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. associate-*r* 99.7%

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

       2. sub-neg 99.7%

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

       3. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 99.7%

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

       1. fma-define 99.7%

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

       2. *-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

       5. associate-*r/ 99.7%

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

       6. +-commutative 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in x around 0 99.1%

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

       1. +-commutative 99.1%

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

       2. fma-define 99.1%

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

       3. associate-*r* 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-*l* 99.1%

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

       6. associate-+r+ 99.2%

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

       7. +-commutative 99.2%

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

       8. associate-*r/ 99.2%

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

       9. +-commutative 99.2%

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

       10. sub-neg 99.2%

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

       11. metadata-eval 99.2%

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

       12. +-commutative 99.2%

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

       13. distribute-rgt-in 99.2%

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

       14. metadata-eval 99.2%

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

       15. metadata-eval 99.2%

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

   17. Simplified 99.2%

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

   if 6.99999999999999989e-6 < x

   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 99.0%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around 0 66.0%

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

       1. sub-neg 66.0%

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

       2. metadata-eval 66.0%

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

       3. fma-define 66.0%

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

       4. sub-neg 66.0%

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

       5. metadata-eval 66.0%

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

       6. associate-*r/ 66.0%

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

       7. metadata-eval 66.0%

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

       8. +-commutative 66.0%

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

   11. Simplified 66.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 1.95 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{t\_0} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))))
   (if (or (<= x -5.8e-6) (not (<= x 1.95e-6)))
     (/
      (+ 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))) (* 6.0 (/ 1.0 t_0)))))
     (/
      (fma -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+ (/ (* (cos y) 6.0) t_0) (+ 1.5 (* (sqrt 5.0) 1.5)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double tmp;
	if ((x <= -5.8e-6) || !(x <= 1.95e-6)) {
		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))) + (6.0 * (1.0 / t_0))));
	} else {
		tmp = fma(-0.0625, (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (((cos(y) * 6.0) / t_0) + (1.5 + (sqrt(5.0) * 1.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if ((x <= -5.8e-6) || !(x <= 1.95e-6))
		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(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(6.0 * Float64(1.0 / t_0)))));
	else
		tmp = Float64(fma(-0.0625, Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(Float64(Float64(cos(y) * 6.0) / t_0) + Float64(1.5 + Float64(sqrt(5.0) * 1.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -5.8e-6], N[Not[LessEqual[x, 1.95e-6]], $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000004e-6 or 1.95e-6 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow1/2 98.8%

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

      4. pow1/2 98.8%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around 0 62.8%

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

   if -5.8000000000000004e-6 < x < 1.95e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. associate-*r* 99.7%

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

       2. sub-neg 99.7%

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

       3. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 99.7%

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

       1. fma-define 99.7%

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

       2. *-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

       5. associate-*r/ 99.7%

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

       6. +-commutative 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in x around 0 99.1%

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

       1. +-commutative 99.1%

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

       2. fma-define 99.1%

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

       3. associate-*r* 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-*l* 99.1%

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

       6. associate-+r+ 99.2%

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

       7. +-commutative 99.2%

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

       8. associate-*r/ 99.2%

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

       9. +-commutative 99.2%

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

       10. sub-neg 99.2%

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

       11. metadata-eval 99.2%

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

       12. +-commutative 99.2%

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

       13. distribute-rgt-in 99.2%

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

       14. metadata-eval 99.2%

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

       15. metadata-eval 99.2%

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

   17. Simplified 99.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 1.95 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{\frac{\cos y \cdot 6}{3 + \sqrt{5}} + \left(1.5 + \sqrt{5} \cdot 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6} \lor \neg \left(x \leq 0.000115\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.8e-6) (not (<= x 0.000115)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
     (+
      1.0
      (+ (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)) (* (- 3.0 (sqrt 5.0)) 0.5)))))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
    (+
     3.0
     (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 1.5 (+ (sqrt 5.0) -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.8e-6) || !(x <= 0.000115)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.8d-6)) .or. (.not. (x <= 0.000115d0))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)) + ((3.0d0 - sqrt(5.0d0)) * 0.5d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (1.5d0 * (sqrt(5.0d0) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.8e-6) || !(x <= 0.000115)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - Math.sqrt(5.0)) * 0.5))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (1.5 * (Math.sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.8e-6) or not (x <= 0.000115):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - math.sqrt(5.0)) * 0.5))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (1.5 * (math.sqrt(5.0) + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.8e-6) || !(x <= 0.000115))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(Float64(3.0 - sqrt(5.0)) * 0.5)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(sqrt(5.0) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.8e-6) || ~((x <= 0.000115)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.8e-6], N[Not[LessEqual[x, 0.000115]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999992e-6 or 1.15e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 62.7%

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

   if -1.79999999999999992e-6 < x < 1.15e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 99.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6} \lor \neg \left(x \leq 0.000115\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_1\right) + 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_0} + 1.5 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -1.6e-5) (not (<= x 5.8e-6)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (+ (* 1.5 (* (cos x) t_1)) (* 6.0 (/ 1.0 t_0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -1.6e-5) || !(x <= 5.8e-6)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_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 = 3.0d0 + sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-1.6d-5)) .or. (.not. (x <= 5.8d-6))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((1.5d0 * (cos(x) * t_1)) + (6.0d0 * (1.0d0 / t_0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / t_0)) + (1.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -1.6e-5) || !(x <= 5.8e-6)) {
		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) * t_1)) + (6.0 * (1.0 / t_0))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / t_0)) + (1.5 * t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -1.6e-5) or not (x <= 5.8e-6):
		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) * t_1)) + (6.0 * (1.0 / t_0))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / t_0)) + (1.5 * t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -1.6e-5) || !(x <= 5.8e-6))
		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(1.5 * Float64(cos(x) * t_1)) + Float64(6.0 * Float64(1.0 / t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -1.6e-5) || ~((x <= 5.8e-6)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.6e-5], N[Not[LessEqual[x, 5.8e-6]], $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999993e-5 or 5.8000000000000004e-6 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow1/2 98.8%

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

      4. pow1/2 98.8%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around 0 62.8%

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

   if -1.59999999999999993e-5 < x < 5.8000000000000004e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 99.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 56.4%

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

10. Final simplification 56.4%

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

Alternative 19: 59.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 56.3%

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

   1. *-commutative 56.3%

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

   2. distribute-lft-out 56.3%

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

7. Simplified 56.3%

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

   1. unpow2 56.3%

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

   2. sin-mult 56.3%

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

9. Applied egg-rr 56.3%

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

    1. div-sub 56.3%

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

    2. +-inverses 56.3%

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

    3. cos-0 56.3%

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

    4. metadata-eval 56.3%

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

    5. count-2 56.3%

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

    6. *-commutative 56.3%

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

11. Simplified 56.3%

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

12. Final simplification 56.3%

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

Alternative 20: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + -1\right)}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (sqrt 2.0)
     (* (- (sin y) (/ (sin x) 16.0)) (- (sin x) (/ (sin y) 16.0))))
    (+ (cos x) -1.0)))
  6.0))

C

double code(double x, double y) {
	return (2.0 + ((sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0)))) * (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 + ((sqrt(2.0d0) * ((sin(y) - (sin(x) / 16.0d0)) * (sin(x) - (sin(y) / 16.0d0)))) * (cos(x) + (-1.0d0)))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.sqrt(2.0) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (Math.cos(x) + -1.0))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + ((math.sqrt(2.0) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sin(x) - (math.sin(y) / 16.0)))) * (math.cos(x) + -1.0))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(cos(x) + -1.0))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0)))) * (cos(x) + -1.0))) / 6.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.2%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.2%

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

   5. associate-+l+ 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 66.8%

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

6. Taylor expanded in y around 0 64.6%

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

7. Taylor expanded in x around 0 41.1%

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

8. Final simplification 41.1%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + -1\right)}{6} \]
9. Add Preprocessing

Alternative 21: 40.1% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 56.3%

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

   1. *-commutative 56.3%

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

   2. distribute-lft-out 56.3%

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

7. Simplified 56.3%

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

8. Taylor expanded in y around 0 32.1%

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

9. Taylor expanded in y around 0 32.0%

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

10. Taylor expanded in y around 0 41.0%

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

11. Final simplification 41.0%

    \[\leadsto 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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))))))