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

Percentage Accurate: 99.3% → 99.3%

Time: 39.6s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * log(exp(((sin(x) - (0.0625d0 * sin(y))) * (sin(y) + (sin(x) * (-0.0625d0))))))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
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.log(Math.exp(((Math.sin(x) - (0.0625 * Math.sin(y))) * (Math.sin(y) + (Math.sin(x) * -0.0625)))))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * math.log(math.exp(((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) + (math.sin(x) * -0.0625)))))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * log(exp(((sin(x) - (0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625)))))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
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[Log[N[Exp[N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   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. fma-def 99.2%

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

   4. associate-+l+ 99.2%

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

   5. distribute-lft-in 99.3%

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

   6. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around -inf 99.4%

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

    1. add-log-exp 99.4%

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

    2. cancel-sign-sub-inv 99.4%

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

    3. metadata-eval 99.4%

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

13. Applied egg-rr 99.4%

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

14. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((sin(y) - (sin(x) * 0.0625d0)) * ((cos(x) - cos(y)) * (sin(x) - (0.0625d0 * sin(y))))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
end function

Java

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

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.cos(x) - math.cos(y)) * (math.sin(x) - (0.0625 * math.sin(y))))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(y) - (sin(x) * 0.0625)) * ((cos(x) - cos(y)) * (sin(x) - (0.0625 * sin(y))))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   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. fma-def 99.2%

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

   4. associate-+l+ 99.2%

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

   5. distribute-lft-in 99.3%

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

   6. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in y around inf 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
end function

Java

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

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))

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(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
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[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   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. fma-def 99.2%

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

   4. associate-+l+ 99.2%

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

   5. distribute-lft-in 99.3%

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

   6. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around -inf 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.104 \lor \neg \left(x \leq 0.122\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + 1.5 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (or (<= x -0.104) (not (<= x 0.122)))
     (/
      (fma
       (* (sqrt 2.0) (sin x))
       (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))
       2.0)
      (+
       3.0
       (+ (* (cos y) (- 4.5 (/ (sqrt 5.0) 0.6666666666666666))) (* 1.5 t_0))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* (sin x) 0.0625)))
         (- (+ 1.0 (* -0.5 (* x x))) (cos y)))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if ((x <= -0.104) || !(x <= 0.122)) {
		tmp = fma((sqrt(2.0) * sin(x)), ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))), 2.0) / (3.0 + ((cos(y) * (4.5 - (sqrt(5.0) / 0.6666666666666666))) + (1.5 * t_0)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if ((x <= -0.104) || !(x <= 0.122))
		tmp = Float64(fma(Float64(sqrt(2.0) * sin(x)), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))), 2.0) / Float64(3.0 + Float64(Float64(cos(y) * Float64(4.5 - Float64(sqrt(5.0) / 0.6666666666666666))) + Float64(1.5 * t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625))) * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.104], N[Not[LessEqual[x, 0.122]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.103999999999999995 or 0.122 < 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. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. fma-udef 99.1%

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

      2. div-sub 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(\cos y \cdot \left(\color{blue}{4.5} - \frac{\sqrt{5}}{0.6666666666666666}\right) + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. div-inv 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + \color{blue}{\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \frac{1}{0.6666666666666666}}\right)} \]

      5. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \color{blue}{1.5}\right)} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)}} \]

   6. Taylor expanded in y around 0 61.4%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \sin x}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   if -0.103999999999999995 < x < 0.122

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
   13. Step-by-step derivation

       1. unpow2 97.8%

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

   14. Simplified 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.104 \lor \neg \left(x \leq 0.122\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \frac{\sqrt{5}}{0.6666666666666666}\right) + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.108 \lor \neg \left(x \leq 0.034\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (or (<= x -0.108) (not (<= x 0.034)))
     (/
      (fma
       (* (sqrt 2.0) (sin x))
       (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))
       2.0)
      (+ 3.0 (* 1.5 (+ t_0 (* (cos y) (- 3.0 (sqrt 5.0)))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* (sin x) 0.0625)))
         (- (+ 1.0 (* -0.5 (* x x))) (cos y)))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if ((x <= -0.108) || !(x <= 0.034)) {
		tmp = fma((sqrt(2.0) * sin(x)), ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * (3.0 - sqrt(5.0))))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if ((x <= -0.108) || !(x <= 0.034))
		tmp = Float64(fma(Float64(sqrt(2.0) * sin(x)), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625))) * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.108], N[Not[LessEqual[x, 0.034]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.107999999999999999 or 0.034000000000000002 < 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. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 61.3%

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

   if -0.107999999999999999 < x < 0.034000000000000002

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
   13. Step-by-step derivation

       1. unpow2 97.8%

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

   14. Simplified 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.108 \lor \neg \left(x \leq 0.034\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.105 \lor \neg \left(x \leq 0.065\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.105) (not (<= x 0.065)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* (sin x) 0.0625)))
         (- (+ 1.0 (* -0.5 (* x x))) (cos y)))))
      (+
       3.0
       (*
        1.5
        (+
         (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
         (* (cos x) (+ (sqrt 5.0) -1.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.105) || !(x <= 0.065)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.105d0)) .or. (.not. (x <= 0.065d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (sin(x) * 0.0625d0))) * ((1.0d0 + ((-0.5d0) * (x * x))) - cos(y))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.105) || !(x <= 0.065)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * (Math.sin(y) - (Math.sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - Math.cos(y))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.105) or not (x <= 0.065):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.cos(x) - math.cos(y)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (math.sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - math.cos(y))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.105) || !(x <= 0.065))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625))) * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.105) || ~((x <= 0.065)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.105], N[Not[LessEqual[x, 0.065]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.104999999999999996 or 0.065000000000000002 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -0.104999999999999996 < x < 0.065000000000000002

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot {x}^{2}\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
   13. Step-by-step derivation

       1. unpow2 97.8%

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

   14. Simplified 99.0%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.105 \lor \neg \left(x \leq 0.065\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0052 \lor \neg \left(x \leq 0.0058\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0052) (not (<= x 0.0058)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* (sin x) 0.0625)))
         (- 1.0 (cos y)))))
      (+
       3.0
       (*
        1.5
        (+
         (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
         (* (cos x) (+ (sqrt 5.0) -1.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0052) || !(x <= 0.0058)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * (1.0 - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0052d0)) .or. (.not. (x <= 0.0058d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (sin(x) * 0.0625d0))) * (1.0d0 - cos(y))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0052) || !(x <= 0.0058)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * (Math.sin(y) - (Math.sin(x) * 0.0625))) * (1.0 - Math.cos(y))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0052) or not (x <= 0.0058):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.cos(x) - math.cos(y)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (math.sin(x) * 0.0625))) * (1.0 - math.cos(y))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0052) || !(x <= 0.0058))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625))) * Float64(1.0 - cos(y))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0052) || ~((x <= 0.0058)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (sqrt(2.0) * (((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))) * (1.0 - cos(y))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0052], N[Not[LessEqual[x, 0.0058]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0051999999999999998 or 0.0058 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -0.0051999999999999998 < x < 0.0058

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 99.1%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0052 \lor \neg \left(x \leq 0.0058\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00016 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot t_1\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_1 \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_2 + \cos y \cdot \left(1.5 - t_2\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00016) (not (<= x 0.0036)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* (- (cos x) (cos y)) t_1)))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (* t_1 (- (+ 1.0 (* -0.5 (* x x))) (cos y)))))
      (* 3.0 (+ 1.0 (- (+ t_2 (* (cos y) (- 1.5 t_2))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00016) || !(x <= 0.0036)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * t_1))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_1 * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 * (1.0 + ((t_2 + (cos(y) * (1.5 - t_2))) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = sin(y) - (sin(x) / 16.0d0)
    t_2 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.00016d0)) .or. (.not. (x <= 0.0036d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((cos(x) - cos(y)) * t_1))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_1 * ((1.0d0 + ((-0.5d0) * (x * x))) - cos(y))))) / (3.0d0 * (1.0d0 + ((t_2 + (cos(y) * (1.5d0 - t_2))) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_2 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00016) || !(x <= 0.0036)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.cos(x) - Math.cos(y)) * t_1))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_1 * ((1.0 + (-0.5 * (x * x))) - Math.cos(y))))) / (3.0 * (1.0 + ((t_2 + (Math.cos(y) * (1.5 - t_2))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.sin(y) - (math.sin(x) / 16.0)
	t_2 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.00016) or not (x <= 0.0036):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.cos(x) - math.cos(y)) * t_1))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_1 * ((1.0 + (-0.5 * (x * x))) - math.cos(y))))) / (3.0 * (1.0 + ((t_2 + (math.cos(y) * (1.5 - t_2))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00016) || !(x <= 0.0036))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(cos(x) - cos(y)) * t_1))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_1 * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_2 + Float64(cos(y) * Float64(1.5 - t_2))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = sin(y) - (sin(x) / 16.0);
	t_2 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.00016) || ~((x <= 0.0036)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * t_1))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_1 * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 * (1.0 + ((t_2 + (cos(y) * (1.5 - t_2))) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.00016], N[Not[LessEqual[x, 0.0036]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000013e-4 or 0.0035999999999999999 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -1.60000000000000013e-4 < x < 0.0035999999999999999

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

      2. associate-+l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. *-commutative 99.5%

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

      7. div-sub 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. unpow2 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00016 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.00096 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (- (cos x) (cos y))))
   (if (or (<= x -0.00096) (not (<= x 0.0036)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* t_1 (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         t_1
         (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* (sin x) 0.0625))))))
      (+
       3.0
       (*
        1.5
        (+ (sqrt 5.0) (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.00096) || !(x <= 0.0036)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_1 * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = cos(x) - cos(y)
    if ((x <= (-0.00096d0)) .or. (.not. (x <= 0.0036d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * (t_1 * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_1 * ((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.00096) || !(x <= 0.0036)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * (t_1 * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_1 * ((Math.sin(x) - (0.0625 * Math.sin(y))) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.00096) or not (x <= 0.0036):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * (t_1 * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_1 * ((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + -1.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.00096) || !(x <= 0.0036))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(t_1 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.00096) || ~((x <= 0.0036)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (sqrt(2.0) * (t_1 * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00096], N[Not[LessEqual[x, 0.0036]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.60000000000000024e-4 or 0.0035999999999999999 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -9.60000000000000024e-4 < x < 0.0035999999999999999

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 98.8%

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

       1. associate--l+ 98.9%

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

   14. Simplified 98.9%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right) + 2}{3 + 1.5 \cdot \color{blue}{\left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{\sqrt{5} + 3} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00096 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \end{array} \]

Alternative 10: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, t_2 \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(t_1 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.00036)
     (/
      (fma (* (sqrt 2.0) (sin x)) (* t_2 (+ (cos x) -1.0)) 2.0)
      (+ 3.0 (* 1.5 (+ t_1 (* (cos y) (- 3.0 (sqrt 5.0)))))))
     (if (<= x 0.0035)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (* t_2 (- (+ 1.0 (* -0.5 (* x x))) (cos y)))))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (/
        (+
         2.0
         (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 (pow (sin x) 2.0)))))
        (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(x) * (sqrt(5.0) + -1.0);
	double t_2 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.00036) {
		tmp = fma((sqrt(2.0) * sin(x)), (t_2 * (cos(x) + -1.0)), 2.0) / (3.0 + (1.5 * (t_1 + (cos(y) * (3.0 - sqrt(5.0))))));
	} else if (x <= 0.0035) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * ((1.0 + (-0.5 * (x * x))) - cos(y))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_1)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.00036)
		tmp = Float64(fma(Float64(sqrt(2.0) * sin(x)), Float64(t_2 * Float64(cos(x) + -1.0)), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_1 + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 0.0035)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_2 * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_1))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00036], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0035], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000023e-4

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      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. fma-def 99.0%

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

      4. associate-+l+ 99.1%

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

      5. distribute-lft-in 99.1%

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

      6. metadata-eval 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in y around 0 60.0%

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

   8. Taylor expanded in y around 0 56.6%

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

   if -3.60000000000000023e-4 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. associate-*l* 99.5%

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

      2. associate-+l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. *-commutative 99.5%

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

      7. div-sub 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. unpow2 98.8%

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

   7. Simplified 98.8%

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

   if 0.00350000000000000007 < 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. fma-def 98.7%

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

      4. associate-+l+ 98.7%

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

      5. distribute-lft-in 98.9%

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

      6. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around -inf 99.1%

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

   12. Taylor expanded in y around 0 59.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \sin x, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;x \leq -0.00088 \lor \neg \left(x \leq 0.0035\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= x -0.00088) (not (<= x 0.0035)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 (pow (sin x) 2.0)))))
      t_0)
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((x <= -0.00088) || !(x <= 0.0035)) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * pow(sin(x), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / 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 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((x <= (-0.00088d0)) .or. (.not. (x <= 0.0035d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * (sin(x) ** 2.0d0))))) / t_0
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((x <= -0.00088) || !(x <= 0.0035)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * Math.pow(Math.sin(x), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (x <= -0.00088) or not (x <= 0.0035):
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * math.pow(math.sin(x), 2.0))))) / t_0
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((x <= -0.00088) || !(x <= 0.0035))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / 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)))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((x <= -0.00088) || ~((x <= 0.0035)))
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * (sin(x) ^ 2.0))))) / t_0;
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00088], N[Not[LessEqual[x, 0.0035]], $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[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.80000000000000031e-4 or 0.00350000000000000007 < 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. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around -inf 99.1%

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

   12. Taylor expanded in y around 0 58.1%

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

   if -8.80000000000000031e-4 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around -inf 99.6%

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

   12. Taylor expanded in x around 0 98.7%

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

       1. associate-*r* 98.7%

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

       2. *-commutative 98.7%

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

   14. Simplified 98.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00088 \lor \neg \left(x \leq 0.0035\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 12: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \mathbf{elif}\;y \leq 0.00085:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_1\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0))))))
        (t_1 (pow (sin y) 2.0)))
   (if (<= y -0.0009)
     (/ (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_0)
     (if (<= y 0.00085)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* -0.0625 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
        t_0)
       (/
        (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_1))))
        t_0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double t_1 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.0009) {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else if (y <= 0.00085) {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_1)))) / 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 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    t_1 = sin(y) ** 2.0d0
    if (y <= (-0.0009d0)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_1 * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / t_0
    else if (y <= 0.00085d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * t_1)))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double t_1 = Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.0009) {
		tmp = (2.0 + (-0.0625 * (t_1 * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / t_0;
	} else if (y <= 0.00085) {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * t_1)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	t_1 = math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.0009:
		tmp = (2.0 + (-0.0625 * (t_1 * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / t_0
	elif y <= 0.00085:
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * t_1)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	t_1 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.0009)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	elseif (y <= 0.00085)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_1)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	t_1 = sin(y) ^ 2.0;
	tmp = 0.0;
	if (y <= -0.0009)
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	elseif (y <= 0.00085)
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_1)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.0009], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 0.00085], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $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$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999998e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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* 99.0%

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

      3. fma-def 98.9%

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

      4. associate-+l+ 99.0%

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

      5. distribute-lft-in 99.1%

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

      6. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in x around -inf 99.2%

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

   12. Taylor expanded in x around 0 60.7%

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

       1. associate-*r* 60.7%

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

       2. *-commutative 60.7%

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

   14. Simplified 60.7%

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

   if -8.9999999999999998e-4 < y < 8.49999999999999953e-4

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around -inf 99.7%

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

   12. Taylor expanded in y around 0 99.7%

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

       1. *-commutative 99.7%

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

       2. associate-*l* 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

   14. Simplified 99.7%

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

   if 8.49999999999999953e-4 < y

   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. fma-def 98.9%

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

      4. associate-+l+ 98.8%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around -inf 99.1%

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

   12. Taylor expanded in x around 0 62.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;y \leq 0.00085:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 13: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := 3 + \sqrt{5}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-6} \lor \neg \left(y \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{t_1} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(t_0 + 4 \cdot \frac{1}{t_1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0))) (t_1 (+ 3.0 (sqrt 5.0))))
   (if (or (<= y -1.7e-6) (not (<= y 6.2e-6)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) t_1)) t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      (+ 3.0 (* 1.5 (+ t_0 (* 4.0 (/ 1.0 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = 3.0 + sqrt(5.0);
	double tmp;
	if ((y <= -1.7e-6) || !(y <= 6.2e-6)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_1)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / 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 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_1 = 3.0d0 + sqrt(5.0d0)
    if ((y <= (-1.7d-6)) .or. (.not. (y <= 6.2d-6))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / t_1)) + t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (3.0d0 + (1.5d0 * (t_0 + (4.0d0 * (1.0d0 / t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_1 = 3.0 + Math.sqrt(5.0);
	double tmp;
	if ((y <= -1.7e-6) || !(y <= 6.2e-6)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / t_1)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_1 = 3.0 + math.sqrt(5.0)
	tmp = 0
	if (y <= -1.7e-6) or not (y <= 6.2e-6):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / t_1)) + t_0)))
	else:
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if ((y <= -1.7e-6) || !(y <= 6.2e-6))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / t_1)) + t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(4.0 * Float64(1.0 / t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (sqrt(5.0) + -1.0);
	t_1 = 3.0 + sqrt(5.0);
	tmp = 0.0;
	if ((y <= -1.7e-6) || ~((y <= 6.2e-6)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_1)) + t_0)));
	else
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -1.7e-6], N[Not[LessEqual[y, 6.2e-6]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(4.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.70000000000000003e-6 or 6.1999999999999999e-6 < y

   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* 99.0%

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

      3. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.1%

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

      6. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around -inf 99.1%

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

   12. Taylor expanded in x around 0 61.5%

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

       1. associate-*r* 61.5%

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

       2. *-commutative 61.5%

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

   14. Simplified 61.5%

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

   if -1.70000000000000003e-6 < y < 6.1999999999999999e-6

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 99.7%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 4 \cdot \frac{1}{\sqrt{5} + 3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-6} \lor \neg \left(y \leq 6.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 14: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;y \leq -0.00074 \lor \neg \left(y \leq 0.00082\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= y -0.00074) (not (<= y 0.00082)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      t_0)
     (/
      (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00074) || !(y <= 0.00082)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((cos(x) + -1.0) * pow(sin(x), 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 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((y <= (-0.00074d0)) .or. (.not. (y <= 0.00082d0))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00074) || !(y <= 0.00082)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (-0.0625 * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (y <= -0.00074) or not (y <= 0.00082):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (-0.0625 * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((y <= -0.00074) || !(y <= 0.00082))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((y <= -0.00074) || ~((y <= 0.00082)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * (-0.0625 * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00074], N[Not[LessEqual[y, 0.00082]], $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] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.3999999999999999e-4 or 8.1999999999999998e-4 < y

   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. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in x around -inf 99.1%

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

   12. Taylor expanded in x around 0 61.2%

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

       1. associate-*r* 61.2%

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

       2. *-commutative 61.2%

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

   14. Simplified 61.2%

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

   if -7.3999999999999999e-4 < y < 8.1999999999999998e-4

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around -inf 99.7%

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

   12. Taylor expanded in y around 0 99.7%

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

       1. *-commutative 99.7%

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

       2. associate-*l* 99.7%

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

       3. sub-neg 99.7%

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

       4. metadata-eval 99.7%

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

   14. Simplified 99.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00074 \lor \neg \left(y \leq 0.00082\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 15: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(\cos x \cdot t_2 + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{t_0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \mathsf{fma}\left(t_2, \cos x, \frac{4}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -4.4e-5)
     (/ t_1 (+ 3.0 (* 1.5 (+ (* (cos x) t_2) (* 4.0 (/ 1.0 t_0))))))
     (if (<= x 0.0035)
       (/
        (+
         2.0
         (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
        (+ 3.0 (* 1.5 (+ -1.0 (+ (sqrt 5.0) (* 4.0 (/ (cos y) t_0)))))))
       (/ t_1 (+ 3.0 (* 1.5 (fma t_2 (cos x) (/ 4.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -4.4e-5) {
		tmp = t_1 / (3.0 + (1.5 * ((cos(x) * t_2) + (4.0 * (1.0 / t_0)))));
	} else if (x <= 0.0035) {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (4.0 * (cos(y) / t_0))))));
	} else {
		tmp = t_1 / (3.0 + (1.5 * fma(t_2, cos(x), (4.0 / t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -4.4e-5)
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_2) + Float64(4.0 * Float64(1.0 / t_0))))));
	elseif (x <= 0.0035)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(-1.0 + Float64(sqrt(5.0) + Float64(4.0 * Float64(cos(y) / t_0)))))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * fma(t_2, cos(x), Float64(4.0 / t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -4.4e-5], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0035], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(-1.0 + N[(N[Sqrt[5.0], $MachinePrecision] + N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3999999999999999e-5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      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. fma-def 99.0%

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

      4. associate-+l+ 99.1%

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

      5. distribute-lft-in 99.1%

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

      6. metadata-eval 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in y around 0 55.1%

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

   if -4.3999999999999999e-5 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around 0 98.7%

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

   if 0.00350000000000000007 < 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. fma-def 98.7%

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

      4. associate-+l+ 98.7%

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

      5. distribute-lft-in 98.9%

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

      6. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in y around 0 57.8%

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

       1. sub-neg 57.8%

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

       2. metadata-eval 57.8%

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

       3. *-commutative 57.8%

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

       4. fma-def 57.8%

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

       5. sub-neg 57.8%

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

       6. metadata-eval 57.8%

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

       7. associate-*r/ 57.8%

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

       8. metadata-eval 57.8%

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

   13. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \frac{4}{\sqrt{5} + 3}\right) \cdot 1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 16: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(\cos x \cdot t_2 + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{t_0} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \mathsf{fma}\left(t_2, \cos x, \frac{4}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -2.2e-5)
     (/ t_1 (+ 3.0 (* 1.5 (+ (* (cos x) t_2) (* 4.0 (/ 1.0 t_0))))))
     (if (<= x 0.0035)
       (/
        (fma -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y)))) 2.0)
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* 4.0 (/ (cos y) t_0)) -1.0)))))
       (/ t_1 (+ 3.0 (* 1.5 (fma t_2 (cos x) (/ 4.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -2.2e-5) {
		tmp = t_1 / (3.0 + (1.5 * ((cos(x) * t_2) + (4.0 * (1.0 / t_0)))));
	} else if (x <= 0.0035) {
		tmp = fma(-0.0625, (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / t_0)) + -1.0))));
	} else {
		tmp = t_1 / (3.0 + (1.5 * fma(t_2, cos(x), (4.0 / t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -2.2e-5)
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_2) + Float64(4.0 * Float64(1.0 / t_0))))));
	elseif (x <= 0.0035)
		tmp = Float64(fma(-0.0625, Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / t_0)) + -1.0)))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * fma(t_2, cos(x), Float64(4.0 / t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.2e-5], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0035], N[(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] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e-5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      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. fma-def 99.0%

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

      4. associate-+l+ 99.1%

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

      5. distribute-lft-in 99.1%

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

      6. metadata-eval 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. sub-neg 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in y around 0 55.1%

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

   if -2.1999999999999999e-5 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around 0 98.7%

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

       1. +-commutative 98.7%

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

       2. fma-def 98.7%

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

       3. associate-*r* 98.7%

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

       4. *-commutative 98.7%

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

       5. associate--l+ 98.7%

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

   13. Simplified 98.7%

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

   if 0.00350000000000000007 < 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. fma-def 98.7%

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

      4. associate-+l+ 98.7%

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

      5. distribute-lft-in 98.9%

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

      6. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in y around 0 57.8%

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

       1. sub-neg 57.8%

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

       2. metadata-eval 57.8%

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

       3. *-commutative 57.8%

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

       4. fma-def 57.8%

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

       5. sub-neg 57.8%

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

       6. metadata-eval 57.8%

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

       7. associate-*r/ 57.8%

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

       8. metadata-eval 57.8%

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

   13. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \frac{4}{\sqrt{5} + 3}\right) \cdot 1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, {\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 17: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-5} \lor \neg \left(x \leq 0.0035\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{t_0}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))))
   (if (or (<= x -4.4e-5) (not (<= x 0.0035)))
     (/
      (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
      (+ 3.0 (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (* 4.0 (/ 1.0 t_0))))))
     (/
      (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
      (+ 3.0 (* 1.5 (+ -1.0 (+ (sqrt 5.0) (* 4.0 (/ (cos y) t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double tmp;
	if ((x <= -4.4e-5) || !(x <= 0.0035)) {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (4.0 * (cos(y) / 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 + sqrt(5.0d0)
    if ((x <= (-4.4d-5)) .or. (.not. (x <= 0.0035d0))) then
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (4.0d0 * (1.0d0 / t_0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (3.0d0 + (1.5d0 * ((-1.0d0) + (sqrt(5.0d0) + (4.0d0 * (cos(y) / 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 tmp;
	if ((x <= -4.4e-5) || !(x <= 0.0035)) {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (Math.sqrt(5.0) + (4.0 * (Math.cos(y) / t_0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	tmp = 0
	if (x <= -4.4e-5) or not (x <= 0.0035):
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))))
	else:
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (math.sqrt(5.0) + (4.0 * (math.cos(y) / t_0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.4e-5) || !(x <= 0.0035))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(4.0 * Float64(1.0 / t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(-1.0 + Float64(sqrt(5.0) + Float64(4.0 * Float64(cos(y) / t_0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	tmp = 0.0;
	if ((x <= -4.4e-5) || ~((x <= 0.0035)))
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	else
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (4.0 * (cos(y) / t_0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.4e-5], N[Not[LessEqual[x, 0.0035]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(-1.0 + N[(N[Sqrt[5.0], $MachinePrecision] + N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999999e-5 or 0.00350000000000000007 < 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. fma-def 98.9%

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

      4. associate-+l+ 98.9%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in y around 0 56.4%

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

   if -4.3999999999999999e-5 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around 0 98.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-5} \lor \neg \left(x \leq 0.0035\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]

Alternative 18: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(t_2 + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(\left(3 + t_2\right) - \sqrt{5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_2 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (<= x -9e-6)
     (* 0.3333333333333333 (/ t_1 (+ 1.0 (* 0.5 (+ t_2 (- 3.0 (sqrt 5.0)))))))
     (if (<= x 0.0035)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
         (+ 0.5 (+ t_0 (* (cos y) (- 1.5 t_0))))))
       (*
        0.3333333333333333
        (/ t_1 (+ 1.0 (* 0.5 (- (+ 3.0 t_2) (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_2 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -9e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - sqrt(5.0))))));
	} else if (x <= 0.0035) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (0.5 + (t_0 + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 + t_2) - 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 = sqrt(5.0d0) * 0.5d0
    t_1 = 2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))
    t_2 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    if (x <= (-9d-6)) then
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * (t_2 + (3.0d0 - sqrt(5.0d0))))))
    else if (x <= 0.0035d0) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (0.5d0 + (t_0 + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * ((3.0d0 + t_2) - sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))));
	double t_2 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -9e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - Math.sqrt(5.0))))));
	} else if (x <= 0.0035) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (0.5 + (t_0 + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 + t_2) - Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = 2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))
	t_2 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	tmp = 0
	if x <= -9e-6:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - math.sqrt(5.0))))))
	elif x <= 0.0035:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (0.5 + (t_0 + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 + t_2) - math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_2 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(t_2 + Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 0.0035)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(0.5 + Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 + t_2) - sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))));
	t_2 = cos(x) * (sqrt(5.0) + -1.0);
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - sqrt(5.0))))));
	elseif (x <= 0.0035)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (0.5 + (t_0 + (cos(y) * (1.5 - t_0)))));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 + t_2) - sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-6], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(t$95$2 + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0035], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(3.0 + t$95$2), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. sub-neg 55.0%

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

      3. metadata-eval 55.0%

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

      4. distribute-lft-out 55.0%

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

      5. *-commutative 55.0%

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

      6. sub-neg 55.0%

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

      7. metadata-eval 55.0%

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

   4. Simplified 55.0%

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

   if -9.00000000000000023e-6 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 98.5%

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

   if 0.00350000000000000007 < 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. Taylor expanded in y around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. sub-neg 57.7%

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

      3. metadata-eval 57.7%

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

      4. distribute-lft-out 57.7%

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

      5. *-commutative 57.7%

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

      6. sub-neg 57.7%

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

      7. metadata-eval 57.7%

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

   4. Simplified 57.7%

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

      1. associate-+r- 57.7%

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

   6. Applied egg-rr 57.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \end{array} \]

Alternative 19: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_1 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + 0.5 \cdot \left(t_1 + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + 0.5 \cdot \left(\left(3 + t_1\right) - \sqrt{5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_1 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (<= x -5e-5)
     (* 0.3333333333333333 (/ t_0 (+ 1.0 (* 0.5 (+ t_1 (- 3.0 (sqrt 5.0)))))))
     (if (<= x 0.0035)
       (/
        (+
         2.0
         (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
        (+
         3.0
         (*
          1.5
          (+ -1.0 (+ (sqrt 5.0) (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))))
       (*
        0.3333333333333333
        (/ t_0 (+ 1.0 (* 0.5 (- (+ 3.0 t_1) (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_1 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -5e-5) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * (t_1 + (3.0 - sqrt(5.0))))));
	} else if (x <= 0.0035) {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (4.0 * (cos(y) / (3.0 + sqrt(5.0))))))));
	} else {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * ((3.0 + t_1) - sqrt(5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))
    t_1 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    if (x <= (-5d-5)) then
        tmp = 0.3333333333333333d0 * (t_0 / (1.0d0 + (0.5d0 * (t_1 + (3.0d0 - sqrt(5.0d0))))))
    else if (x <= 0.0035d0) then
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (3.0d0 + (1.5d0 * ((-1.0d0) + (sqrt(5.0d0) + (4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0))))))))
    else
        tmp = 0.3333333333333333d0 * (t_0 / (1.0d0 + (0.5d0 * ((3.0d0 + t_1) - sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))));
	double t_1 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -5e-5) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * (t_1 + (3.0 - Math.sqrt(5.0))))));
	} else if (x <= 0.0035) {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (Math.sqrt(5.0) + (4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0))))))));
	} else {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * ((3.0 + t_1) - Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))
	t_1 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	tmp = 0
	if x <= -5e-5:
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * (t_1 + (3.0 - math.sqrt(5.0))))))
	elif x <= 0.0035:
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (3.0 + (1.5 * (-1.0 + (math.sqrt(5.0) + (4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))))))
	else:
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * ((3.0 + t_1) - math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_1 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if (x <= -5e-5)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 0.0035)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(-1.0 + Float64(sqrt(5.0) + Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 + t_1) - sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))));
	t_1 = cos(x) * (sqrt(5.0) + -1.0);
	tmp = 0.0;
	if (x <= -5e-5)
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * (t_1 + (3.0 - sqrt(5.0))))));
	elseif (x <= 0.0035)
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (3.0 + (1.5 * (-1.0 + (sqrt(5.0) + (4.0 * (cos(y) / (3.0 + sqrt(5.0))))))));
	else
		tmp = 0.3333333333333333 * (t_0 / (1.0 + (0.5 * ((3.0 + t_1) - sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-5], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0035], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(-1.0 + N[(N[Sqrt[5.0], $MachinePrecision] + N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(0.5 * N[(N[(3.0 + t$95$1), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000024e-5

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. sub-neg 55.0%

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

      3. metadata-eval 55.0%

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

      4. distribute-lft-out 55.0%

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

      5. *-commutative 55.0%

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

      6. sub-neg 55.0%

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

      7. metadata-eval 55.0%

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

   4. Simplified 55.0%

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

   if -5.00000000000000024e-5 < x < 0.00350000000000000007

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. associate-+l+ 99.5%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + 1.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around 0 98.7%

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

   if 0.00350000000000000007 < 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. Taylor expanded in y around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. sub-neg 57.7%

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

      3. metadata-eval 57.7%

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

      4. distribute-lft-out 57.7%

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

      5. *-commutative 57.7%

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

      6. sub-neg 57.7%

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

      7. metadata-eval 57.7%

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

   4. Simplified 57.7%

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

      1. associate-+r- 57.7%

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

   6. Applied egg-rr 57.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(-1 + \left(\sqrt{5} + 4 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \end{array} \]

Alternative 20: 60.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((sin(x) ^ 2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot {\sin x}^{2}\right)}\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. distribute-lft-out 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}} \]

   5. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   6. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

4. Simplified 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

5. Final simplification 55.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)} \]

Alternative 21: 60.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
   (+ 1.0 (* 0.5 (- (+ 3.0 (* (cos x) (+ (sqrt 5.0) -1.0))) (sqrt 5.0)))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (1.0d0 + (0.5d0 * ((3.0d0 + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) - sqrt(5.0d0)))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) - Math.sqrt(5.0)))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 + (math.cos(x) * (math.sqrt(5.0) + -1.0))) - math.sqrt(5.0)))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) - sqrt(5.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (1.0 + (0.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0)))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot {\sin x}^{2}\right)}\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. distribute-lft-out 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}} \]

   5. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   6. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

4. Simplified 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-+r- 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{5} + -1\right) + 3\right) - \sqrt{5}\right)}} \]

6. Applied egg-rr 55.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{5} + -1\right) + 3\right) - \sqrt{5}\right)}} \]

7. Final simplification 55.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)} \]

Alternative 22: 40.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
   2.0)))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / 2.0);
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / 2.0)

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / 2.0);
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot {\sin x}^{2}\right)}\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. distribute-lft-out 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}} \]

   5. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   6. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

4. Simplified 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0 37.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \color{blue}{2}} \]

6. Final simplification 37.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{2} \]

Alternative 23: 40.3% accurate, 1139.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot {\sin x}^{2}\right)}\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. distribute-lft-out 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}} \]

   5. *-commutative 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   6. sub-neg 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)} \]

   7. metadata-eval 55.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)} \]

4. Simplified 55.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0 37.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \color{blue}{2}} \]

6. Taylor expanded in x around 0 29.0%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(-0.5 \cdot \left(\sqrt{2} \cdot {x}^{4}\right)\right)}}{1 + 0.5 \cdot 2} \]
7. Step-by-step derivation

   1. *-commutative 29.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(-0.5 \cdot \color{blue}{\left({x}^{4} \cdot \sqrt{2}\right)}\right)}{1 + 0.5 \cdot 2} \]

   2. associate-*r* 29.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(\left(-0.5 \cdot {x}^{4}\right) \cdot \sqrt{2}\right)}}{1 + 0.5 \cdot 2} \]

8. Simplified 29.0%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(\left(-0.5 \cdot {x}^{4}\right) \cdot \sqrt{2}\right)}}{1 + 0.5 \cdot 2} \]

9. Taylor expanded in x around 0 37.2%

   \[\leadsto \color{blue}{0.3333333333333333} \]

10. Final simplification 37.2%

    \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023230 
(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))))))