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

Percentage Accurate: 99.3% → 99.3%

Time: 41.5s

Alternatives: 18

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{4}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (sin x) (/ (sin y) 16.0))
   (* (sqrt 2.0) (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y))))
   2.0)
  (fma
   3.0
   (fma (cos x) (+ (/ (sqrt 5.0) 2.0) -0.5) 1.0)
   (/ (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))) 0.6666666666666666))))

C

double code(double x, double y) {
	return fma((sin(x) - (sin(y) / 16.0)), (sqrt(2.0) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y)))), 2.0) / fma(3.0, fma(cos(x), ((sqrt(5.0) / 2.0) + -0.5), 1.0), ((cos(y) * (4.0 / (3.0 + sqrt(5.0)))) / 0.6666666666666666));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sin(x) - Float64(sin(y) / 16.0)), Float64(sqrt(2.0) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / fma(3.0, fma(cos(x), Float64(Float64(sqrt(5.0) / 2.0) + -0.5), 1.0), Float64(Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))) / 0.6666666666666666)))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   3. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   4. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

7. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{4}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (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) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625d0)) * (sin(y) - (sin(x) * 0.0625d0)))))) / ((3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))) + (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.cos(x) - Math.cos(y)) * ((Math.sin(x) - (Math.sin(y) * 0.0625)) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / ((3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (math.sin(y) * 0.0625)) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / ((3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))) + (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(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)))) + 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) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   3. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   4. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

7. Taylor expanded in x around inf 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
9. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(x) + (sin(y) * -0.0625)) * (sin(y) + (sin(x) * -0.0625))))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (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) * (cos(x) - cos(y))) * ((sin(x) + (sin(y) * (-0.0625d0))) * (sin(y) + (sin(x) * (-0.0625d0)))))) / ((3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))) + (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.cos(x) - Math.cos(y))) * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * (Math.sin(y) + (Math.sin(x) * -0.0625))))) / ((3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return (2.0 + ((math.sqrt(2.0) * (math.cos(x) - math.cos(y))) * ((math.sin(x) + (math.sin(y) * -0.0625)) * (math.sin(y) + (math.sin(x) * -0.0625))))) / ((3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(sin(y) + Float64(sin(x) * -0.0625))))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)))) + 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) * (cos(x) - cos(y))) * ((sin(x) + (sin(y) * -0.0625)) * (sin(y) + (sin(x) * -0.0625))))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   3. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   4. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

7. Taylor expanded in x around inf 99.4%

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

   1. add0 99.4%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 0\right)}}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   2. associate-*r* 99.3%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)} + 0\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   3. cancel-sign-sub-inv 99.3%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \color{blue}{\left(\sin y + \left(-0.0625\right) \cdot \sin x\right)}\right) + 0\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y + \color{blue}{-0.0625} \cdot \sin x\right)\right) + 0\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

9. Applied egg-rr 99.3%

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) + 0\right)}}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
10. Step-by-step derivation

    1. add0 99.3%

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

    2. associate-*l* 99.4%

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

    3. associate-*r* 99.4%

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

    4. cancel-sign-sub-inv 99.4%

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

    5. metadata-eval 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
13. Add Preprocessing

Alternative 5: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0046 \lor \neg \left(x \leq 0.00156\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.0046) (not (<= x 0.00156)))
   (/
    (+
     2.0
     (*
      (- (cos x) (cos y))
      (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))))
    (*
     3.0
     (+
      (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
      (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
   (/
    (+
     2.0
     (*
      (sqrt 2.0)
      (*
       (- 1.0 (cos y))
       (fma -0.0625 (pow (sin y) 2.0) (* x (* (sin y) 1.00390625))))))
    (+
     (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))
     (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.0046) || !(x <= 0.00156)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))))) / (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 + (sqrt(2.0) * ((1.0 - cos(y)) * fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) * 1.00390625)))))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.0046) || !(x <= 0.00156))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))))) / 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)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.0046], N[Not[LessEqual[x, 0.00156]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0045999999999999999 or 0.00155999999999999997 < 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. Add Preprocessing

   3. Taylor expanded in y around 0 64.8%

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

      1. *-commutative 64.8%

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

   5. Simplified 64.8%

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

   if -0.0045999999999999999 < x < 0.00155999999999999997

   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. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 99.8%

      \[\leadsto \frac{2 + \sqrt{2} \cdot \color{blue}{\left(-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right) + x \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
   9. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)} + x \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      2. associate-*r* 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right) + \color{blue}{\left(x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right) \cdot \left(1 - \cos y\right)}\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. distribute-rgt-out 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)}}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. fma-define 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. distribute-rgt1-in 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\color{blue}{1.00390625} \cdot \sin y\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   10. Simplified 99.8%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(1.00390625 \cdot \sin y\right)\right)\right)}}{3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0046 \lor \neg \left(x \leq 0.00156\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0072 \lor \neg \left(x \leq 0.00156\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_0 \cdot \left(\sin x \cdot \sqrt{2}\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

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 -0.0072) (not (<= x 0.00156)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sin x) (sqrt 2.0))))) t_1)
     (/
      (+
       2.0
       (* (- 1.0 (cos y)) (* t_0 (* (sqrt 2.0) (+ x (* (sin y) -0.0625))))))
      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 <= -0.0072) || !(x <= 0.00156)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sin(x) * sqrt(2.0))))) / t_1;
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / 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 <= (-0.0072d0)) .or. (.not. (x <= 0.00156d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_0 * (sin(x) * sqrt(2.0d0))))) / t_1
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_0 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / 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 <= -0.0072) || !(x <= 0.00156)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_0 * (Math.sin(x) * Math.sqrt(2.0))))) / t_1;
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_0 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / 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 <= -0.0072) or not (x <= 0.00156):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_0 * (math.sin(x) * math.sqrt(2.0))))) / t_1
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_0 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / 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 <= -0.0072) || !(x <= 0.00156))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_0 * Float64(sin(x) * sqrt(2.0))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / 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 <= -0.0072) || ~((x <= 0.00156)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sin(x) * sqrt(2.0))))) / t_1;
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / 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, -0.0072], N[Not[LessEqual[x, 0.00156]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0071999999999999998 or 0.00155999999999999997 < 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. Add Preprocessing

   3. Taylor expanded in y around 0 64.8%

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

      1. *-commutative 64.8%

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

   5. Simplified 64.8%

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

   if -0.0071999999999999998 < x < 0.00155999999999999997

   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%

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

      1. associate-*r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-out 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-lft-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0072 \lor \neg \left(x \leq 0.00156\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0058:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot t\_0\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.00156:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (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 (<= x -0.0058)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (+ 0.0625 (* (cos x) -0.0625)) t_0)))
      (+
       (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))
       (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
     (if (<= x 0.00156)
       (/
        (+
         2.0
         (*
          (- 1.0 (cos y))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (+ x (* (sin y) -0.0625))))))
        t_1)
       (/
        (+ 2.0 (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) t_0))))
        t_1)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	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 <= -0.0058) {
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * t_0))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	} else if (x <= 0.00156) {
		tmp = (2.0 + ((1.0 - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * t_0)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    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 <= (-0.0058d0)) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((0.0625d0 + (cos(x) * (-0.0625d0))) * t_0))) / ((3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))) + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))
    else if (x <= 0.00156d0) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / t_1
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * t_0)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = 3.0 * ((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 <= -0.0058) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((0.0625 + (Math.cos(x) * -0.0625)) * t_0))) / ((3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))));
	} else if (x <= 0.00156) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / t_1;
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * t_0)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	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 <= -0.0058:
		tmp = (2.0 + (math.sqrt(2.0) * ((0.0625 + (math.cos(x) * -0.0625)) * t_0))) / ((3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))
	elif x <= 0.00156:
		tmp = (2.0 + ((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / t_1
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * t_0)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	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 <= -0.0058)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(0.0625 + Float64(cos(x) * -0.0625)) * t_0))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	elseif (x <= 0.00156)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * t_0)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	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 <= -0.0058)
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * t_0))) / ((3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	elseif (x <= 0.00156)
		tmp = (2.0 + ((1.0 - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * t_0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(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[LessEqual[x, -0.0058], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.0625 + N[(N[Cos[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00156], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0058

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 98.8%

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

      5. fma-define 98.8%

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

      6. distribute-lft-in 98.7%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 98.8%

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

   8. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.3%

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

      2. associate-*r* 65.3%

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

      3. sub-neg 65.3%

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

      4. metadata-eval 65.3%

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

      5. distribute-lft-in 65.3%

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

      6. metadata-eval 65.3%

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

   10. Simplified 65.4%

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

   if -0.0058 < x < 0.00155999999999999997

   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%

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

      1. associate-*r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-out 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-lft-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   if 0.00155999999999999997 < 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 56.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0058:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.00156:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\\ \mathbf{if}\;y \leq -5.3 \cdot 10^{-6} \lor \neg \left(y \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{t\_1 + 6 \cdot \frac{\cos y}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{t\_1 + 6 \cdot \frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))))
   (if (or (<= y -5.3e-6) (not (<= y 2.6e-6)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 (pow (sin y) 2.0)))))
      (+ t_1 (* 6.0 (/ (cos y) t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ t_1 (* 6.0 (/ 1.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((y <= -5.3e-6) || !(y <= 2.6e-6)) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * pow(sin(y), 2.0))))) / (t_1 + (6.0 * (cos(y) / t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = 3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))
    if ((y <= (-5.3d-6)) .or. (.not. (y <= 2.6d-6))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (t_1 + (6.0d0 * (cos(y) / t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (t_1 + (6.0d0 * (1.0d0 / t_0)))
    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 = 3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((y <= -5.3e-6) || !(y <= 2.6e-6)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (t_1 + (6.0 * (Math.cos(y) / t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = 3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))
	tmp = 0
	if (y <= -5.3e-6) or not (y <= 2.6e-6):
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (t_1 + (6.0 * (math.cos(y) / t_0)))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5))))
	tmp = 0.0
	if ((y <= -5.3e-6) || !(y <= 2.6e-6))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(t_1 + Float64(6.0 * Float64(cos(y) / t_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(t_1 + Float64(6.0 * Float64(1.0 / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	tmp = 0.0;
	if ((y <= -5.3e-6) || ~((y <= 2.6e-6)))
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * (sin(y) ^ 2.0))))) / (t_1 + (6.0 * (cos(y) / t_0)));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -5.3e-6], N[Not[LessEqual[y, 2.6e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $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[(t$95$1 + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3000000000000001e-6 or 2.60000000000000009e-6 < y

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l* 99.1%

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

      5. fma-define 99.1%

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

      6. distribute-lft-in 99.2%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.3%

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

   8. Taylor expanded in x around 0 62.9%

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

   if -5.3000000000000001e-6 < y < 2.60000000000000009e-6

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

      5. fma-define 99.4%

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

      6. distribute-lft-in 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in y around 0 98.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{-6} \lor \neg \left(y \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\\ t_2 := t\_1 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot t\_0\right)}{t\_2}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t\_0\right)\right)}{t\_1 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)))))
        (t_2 (+ t_1 (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))
   (if (<= x -0.0012)
     (/ (+ 2.0 (* (sqrt 2.0) (* (+ 0.0625 (* (cos x) -0.0625)) t_0))) t_2)
     (if (<= x 0.001)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
        t_2)
       (/
        (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_0))))
        (+ t_1 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	double t_2 = t_1 + (6.0 * (cos(y) / (3.0 + sqrt(5.0))));
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * t_0))) / t_2;
	} else if (x <= 0.001) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / t_2;
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_0)))) / (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = 3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))
    t_2 = t_1 + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0))))
    if (x <= (-0.0012d0)) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((0.0625d0 + (cos(x) * (-0.0625d0))) * t_0))) / t_2
    else if (x <= 0.001d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / t_2
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * t_0)))) / (t_1 + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = 3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)));
	double t_2 = t_1 + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0))));
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((0.0625 + (Math.cos(x) * -0.0625)) * t_0))) / t_2;
	} else if (x <= 0.001) {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / t_2;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * t_0)))) / (t_1 + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = 3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))
	t_2 = t_1 + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))
	tmp = 0
	if x <= -0.0012:
		tmp = (2.0 + (math.sqrt(2.0) * ((0.0625 + (math.cos(x) * -0.0625)) * t_0))) / t_2
	elif x <= 0.001:
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / t_2
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * t_0)))) / (t_1 + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5))))
	t_2 = Float64(t_1 + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))
	tmp = 0.0
	if (x <= -0.0012)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(0.0625 + Float64(cos(x) * -0.0625)) * t_0))) / t_2);
	elseif (x <= 0.001)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / t_2);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_0)))) / Float64(t_1 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	t_2 = t_1 + (6.0 * (cos(y) / (3.0 + sqrt(5.0))));
	tmp = 0.0;
	if (x <= -0.0012)
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * t_0))) / t_2;
	elseif (x <= 0.001)
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / t_2;
	else
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_0)))) / (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0012], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.0625 + N[(N[Cos[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.001], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00119999999999999989

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 98.8%

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

      5. fma-define 98.8%

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

      6. distribute-lft-in 98.7%

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

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 98.8%

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

   8. Taylor expanded in y around 0 65.4%

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

      1. *-commutative 65.3%

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

      2. associate-*r* 65.3%

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

      3. sub-neg 65.3%

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

      4. metadata-eval 65.3%

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

      5. distribute-lft-in 65.3%

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

      6. metadata-eval 65.3%

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

   10. Simplified 65.4%

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

   if -0.00119999999999999989 < x < 1e-3

   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. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 99.2%

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

   if 1e-3 < 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. +-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. fma-define 98.9%

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

      6. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in y around 0 56.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{-6} \lor \neg \left(y \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t\_1 + 6 \cdot \frac{\cos y}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{t\_1 + 6 \cdot \frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))))
   (if (or (<= y -7.1e-6) (not (<= y 2.5e-6)))
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
      (+ t_1 (* 6.0 (/ (cos y) t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ t_1 (* 6.0 (/ 1.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((y <= -7.1e-6) || !(y <= 2.5e-6)) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (t_1 + (6.0 * (cos(y) / t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = 3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))
    if ((y <= (-7.1d-6)) .or. (.not. (y <= 2.5d-6))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (t_1 + (6.0d0 * (cos(y) / t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (t_1 + (6.0d0 * (1.0d0 / t_0)))
    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 = 3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((y <= -7.1e-6) || !(y <= 2.5e-6)) {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (t_1 + (6.0 * (Math.cos(y) / t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = 3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))
	tmp = 0
	if (y <= -7.1e-6) or not (y <= 2.5e-6):
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (t_1 + (6.0 * (math.cos(y) / t_0)))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5))))
	tmp = 0.0
	if ((y <= -7.1e-6) || !(y <= 2.5e-6))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(t_1 + Float64(6.0 * Float64(cos(y) / t_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(t_1 + Float64(6.0 * Float64(1.0 / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	tmp = 0.0;
	if ((y <= -7.1e-6) || ~((y <= 2.5e-6)))
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (t_1 + (6.0 * (cos(y) / t_0)));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (t_1 + (6.0 * (1.0 / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -7.1e-6], N[Not[LessEqual[y, 2.5e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $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[(t$95$1 + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0999999999999998e-6 or 2.5000000000000002e-6 < y

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-*l* 99.1%

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

      5. fma-define 99.1%

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

      6. distribute-lft-in 99.2%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.3%

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

   8. Taylor expanded in x around 0 62.7%

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

   if -7.0999999999999998e-6 < y < 2.5000000000000002e-6

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

      5. fma-define 99.4%

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

      6. distribute-lft-in 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in y around 0 98.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-6} \lor \neg \left(y \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\\ \mathbf{if}\;x \leq -0.0013 \lor \neg \left(x \leq 0.00092\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{t\_0 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t\_0 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))))
   (if (or (<= x -0.0013) (not (<= x 0.00092)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ 0.0625 (* (cos x) -0.0625)) (pow (sin x) 2.0))))
      (+ t_0 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))))
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
      (+ t_0 (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((x <= -0.0013) || !(x <= 0.00092)) {
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * pow(sin(x), 2.0)))) / (t_0 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (t_0 + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))
    if ((x <= (-0.0013d0)) .or. (.not. (x <= 0.00092d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((0.0625d0 + (cos(x) * (-0.0625d0))) * (sin(x) ** 2.0d0)))) / (t_0 + (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (t_0 + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (1.0 + (Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)));
	double tmp;
	if ((x <= -0.0013) || !(x <= 0.00092)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((0.0625 + (Math.cos(x) * -0.0625)) * Math.pow(Math.sin(x), 2.0)))) / (t_0 + (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (t_0 + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))
	tmp = 0
	if (x <= -0.0013) or not (x <= 0.00092):
		tmp = (2.0 + (math.sqrt(2.0) * ((0.0625 + (math.cos(x) * -0.0625)) * math.pow(math.sin(x), 2.0)))) / (t_0 + (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (t_0 + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5))))
	tmp = 0.0
	if ((x <= -0.0013) || !(x <= 0.00092))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(0.0625 + Float64(cos(x) * -0.0625)) * (sin(x) ^ 2.0)))) / Float64(t_0 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(t_0 + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)));
	tmp = 0.0;
	if ((x <= -0.0013) || ~((x <= 0.00092)))
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * (sin(x) ^ 2.0)))) / (t_0 + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	else
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (t_0 + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0013], N[Not[LessEqual[x, 0.00092]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.0625 + N[(N[Cos[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0012999999999999999 or 9.2000000000000003e-4 < 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. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 98.9%

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

      5. fma-define 98.9%

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

      6. distribute-lft-in 98.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.9%

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

   6. Taylor expanded in y around 0 61.3%

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

      1. *-commutative 61.3%

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

      2. associate-*r* 61.3%

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

      3. sub-neg 61.3%

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

      4. metadata-eval 61.3%

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

      5. distribute-lft-in 61.3%

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

      6. metadata-eval 61.3%

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

   8. Simplified 61.3%

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

   if -0.0012999999999999999 < x < 9.2000000000000003e-4

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 99.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0013 \lor \neg \left(x \leq 0.00092\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.00132 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (* 3.0 (+ 1.0 (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))
          (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))
   (if (or (<= x -0.00132) (not (<= x 0.0009)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (+ 0.0625 (* (cos x) -0.0625)) (pow (sin x) 2.0))))
      t_0)
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))));
	double tmp;
	if ((x <= -0.00132) || !(x <= 0.0009)) {
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * pow(sin(x), 2.0)))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (3.0d0 * (1.0d0 + (cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)))) + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0))))
    if ((x <= (-0.00132d0)) .or. (.not. (x <= 0.0009d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((0.0625d0 + (cos(x) * (-0.0625d0))) * (sin(x) ** 2.0d0)))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / 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) * 0.5) - 0.5)))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0))));
	double tmp;
	if ((x <= -0.00132) || !(x <= 0.0009)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((0.0625 + (Math.cos(x) * -0.0625)) * Math.pow(Math.sin(x), 2.0)))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (3.0 * (1.0 + (math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))
	tmp = 0
	if (x <= -0.00132) or not (x <= 0.0009):
		tmp = (2.0 + (math.sqrt(2.0) * ((0.0625 + (math.cos(x) * -0.0625)) * math.pow(math.sin(x), 2.0)))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))
	tmp = 0.0
	if ((x <= -0.00132) || !(x <= 0.0009))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(0.0625 + Float64(cos(x) * -0.0625)) * (sin(x) ^ 2.0)))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (3.0 * (1.0 + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))));
	tmp = 0.0;
	if ((x <= -0.00132) || ~((x <= 0.0009)))
		tmp = (2.0 + (sqrt(2.0) * ((0.0625 + (cos(x) * -0.0625)) * (sin(x) ^ 2.0)))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00132], N[Not[LessEqual[x, 0.0009]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.0625 + N[(N[Cos[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00132 or 8.9999999999999998e-4 < 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. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 98.9%

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

      5. fma-define 98.9%

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

      6. distribute-lft-in 98.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.0%

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

   8. Taylor expanded in y around 0 61.4%

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

      1. *-commutative 61.3%

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

      2. associate-*r* 61.3%

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

      3. sub-neg 61.3%

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

      4. metadata-eval 61.3%

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

      5. distribute-lft-in 61.3%

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

      6. metadata-eval 61.3%

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

   10. Simplified 61.4%

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

   if -0.00132 < x < 8.9999999999999998e-4

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 99.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00132 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(0.0625 + \cos x \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -4 \cdot 10^{-7} \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(t\_1 - 0.5\right)\right) + 6 \cdot \frac{1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{t\_0} + 3 \cdot \left(0.5 + t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))) (t_1 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -4e-7) (not (<= x 0.000145)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ (* 3.0 (+ 1.0 (* (cos x) (- t_1 0.5)))) (* 6.0 (/ 1.0 t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ (* 6.0 (/ (cos y) t_0)) (* 3.0 (+ 0.5 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -4e-7) || !(x <= 0.000145)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((3.0 * (1.0 + (cos(x) * (t_1 - 0.5)))) + (6.0 * (1.0 / t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / t_0)) + (3.0 * (0.5 + t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-4d-7)) .or. (.not. (x <= 0.000145d0))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((3.0d0 * (1.0d0 + (cos(x) * (t_1 - 0.5d0)))) + (6.0d0 * (1.0d0 / t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / ((6.0d0 * (cos(y) / t_0)) + (3.0d0 * (0.5d0 + t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -4e-7) || !(x <= 0.000145)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((3.0 * (1.0 + (Math.cos(x) * (t_1 - 0.5)))) + (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)))))) / ((6.0 * (Math.cos(y) / t_0)) + (3.0 * (0.5 + t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -4e-7) or not (x <= 0.000145):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((3.0 * (1.0 + (math.cos(x) * (t_1 - 0.5)))) + (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)))))) / ((6.0 * (math.cos(y) / t_0)) + (3.0 * (0.5 + t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -4e-7) || !(x <= 0.000145))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(t_1 - 0.5)))) + 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(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(3.0 * Float64(0.5 + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -4e-7) || ~((x <= 0.000145)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((3.0 * (1.0 + (cos(x) * (t_1 - 0.5)))) + (6.0 * (1.0 / t_0)));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / t_0)) + (3.0 * (0.5 + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -4e-7], N[Not[LessEqual[x, 0.000145]], $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[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $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[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999998e-7 or 1.45e-4 < x

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. fma-define 98.9%

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

      6. distribute-lft-in 98.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in y around 0 61.2%

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

   if -3.9999999999999998e-7 < x < 1.45e-4

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around 0 99.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7} \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -4 \cdot 10^{-7} \lor \neg \left(x \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(t\_0 - 0.5\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(0.5 + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -4e-7) (not (<= x 5.8e-5)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ (* 3.0 (+ 1.0 (* (cos x) (- t_0 0.5)))) (* 1.5 (- 3.0 (sqrt 5.0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 3.0 (+ 0.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -4e-7) || !(x <= 5.8e-5)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((3.0 * (1.0 + (cos(x) * (t_0 - 0.5)))) + (1.5 * (3.0 - sqrt(5.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (3.0 * (0.5 + t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-4d-7)) .or. (.not. (x <= 5.8d-5))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((3.0d0 * (1.0d0 + (cos(x) * (t_0 - 0.5d0)))) + (1.5d0 * (3.0d0 - sqrt(5.0d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (3.0d0 * (0.5d0 + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -4e-7) || !(x <= 5.8e-5)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((3.0 * (1.0 + (Math.cos(x) * (t_0 - 0.5)))) + (1.5 * (3.0 - Math.sqrt(5.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (3.0 * (0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -4e-7) or not (x <= 5.8e-5):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((3.0 * (1.0 + (math.cos(x) * (t_0 - 0.5)))) + (1.5 * (3.0 - math.sqrt(5.0))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (3.0 * (0.5 + t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -4e-7) || !(x <= 5.8e-5))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5)))) + Float64(1.5 * Float64(3.0 - sqrt(5.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(3.0 * Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -4e-7) || ~((x <= 5.8e-5)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((3.0 * (1.0 + (cos(x) * (t_0 - 0.5)))) + (1.5 * (3.0 - sqrt(5.0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (3.0 * (0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -4e-7], N[Not[LessEqual[x, 5.8e-5]], $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[(N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $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[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999998e-7 or 5.8e-5 < 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. +-commutative 98.9%

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

      2. associate-*l* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. fma-define 98.9%

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

      6. distribute-lft-in 98.9%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 61.1%

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

   if -3.9999999999999998e-7 < x < 5.8e-5

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. fma-define 99.7%

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

      6. distribute-lft-in 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   7. Taylor expanded in x around 0 99.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7} \lor \neg \left(x \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 58.6%

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

6. Final simplification 58.6%

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

Alternative 16: 59.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (3.0 * (0.5 + (sqrt(5.0) * 0.5))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (3.0d0 * (0.5d0 + (sqrt(5.0d0) * 0.5d0))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (3.0 * (0.5 + (Math.sqrt(5.0) * 0.5))));
}

Python

def code(x, y):
	return (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (3.0 * (0.5 + (math.sqrt(5.0) * 0.5))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(3.0 * Float64(0.5 + Float64(sqrt(5.0) * 0.5)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (3.0 * (0.5 + (sqrt(5.0) * 0.5))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   3. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   4. pow1/2 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}\right)} \]

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}\right)} \]

7. Taylor expanded in x around 0 58.6%

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

8. Final simplification 58.6%

   \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 17: 42.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.3%

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

6. Taylor expanded in y around 0 60.5%

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

7. Taylor expanded in x around 0 39.5%

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

8. Final simplification 39.5%

   \[\leadsto \frac{2}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 18: 40.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (+ (* 3.0 (+ 0.5 (* (sqrt 5.0) 0.5))) (* 1.5 (- 3.0 (sqrt 5.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(3.0 * N[(0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*l* 99.2%

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

   5. fma-define 99.2%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sin x - \frac{\sin y}{16}, \sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, 1\right), \frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{0.6666666666666666}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 58.6%

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

6. Taylor expanded in y around 0 27.2%

   \[\leadsto \frac{2 + -0.0625 \cdot \color{blue}{\left(0.5 \cdot \left({y}^{4} \cdot \sqrt{2}\right)\right)}}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]
7. Step-by-step derivation

   1. *-commutative 27.2%

      \[\leadsto \frac{2 + -0.0625 \cdot \color{blue}{\left(\left({y}^{4} \cdot \sqrt{2}\right) \cdot 0.5\right)}}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]

   2. *-commutative 27.2%

      \[\leadsto \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sqrt{2} \cdot {y}^{4}\right)} \cdot 0.5\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]

   3. associate-*l* 27.2%

      \[\leadsto \frac{2 + -0.0625 \cdot \color{blue}{\left(\sqrt{2} \cdot \left({y}^{4} \cdot 0.5\right)\right)}}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]

8. Simplified 27.2%

   \[\leadsto \frac{2 + -0.0625 \cdot \color{blue}{\left(\sqrt{2} \cdot \left({y}^{4} \cdot 0.5\right)\right)}}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]

9. Taylor expanded in y around 0 26.9%

   \[\leadsto \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({y}^{4} \cdot 0.5\right)\right)}{\color{blue}{1.5 \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)} \]

10. Taylor expanded in y around 0 37.3%

    \[\leadsto \color{blue}{\frac{2}{1.5 \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)}} \]

11. Final simplification 37.3%

    \[\leadsto \frac{2}{3 \cdot \left(0.5 + \sqrt{5} \cdot 0.5\right) + 1.5 \cdot \left(3 - \sqrt{5}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))))))