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

Percentage Accurate: 99.3% → 99.4%

Time: 42.0s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (- (sin y) (/ (sin x) 16.0))
    (* (- (sin x) (/ (sin y) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (+
    (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))
    (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0)))))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.3%

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

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25} - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \color{blue}{1.5}\right) \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{1.5}\right)}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot \color{blue}{1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
6. Step-by-step derivation

   1. swap-sqr 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot 1.5\right)}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   2. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{5} \cdot \left(1.5 \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - 5 \cdot \color{blue}{2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. cancel-sign-sub-inv 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25 + \left(-5\right) \cdot 2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-11.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{9}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   8. +-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\sqrt{5} \cdot 1.5 + 4.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   9. fma-def 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

7. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

8. Taylor expanded in y around inf 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}} \]

9. Final simplification 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. add-sqr-sqrt 99.4%

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

   4. metadata-eval 99.4%

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

3. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around -inf 99.3%

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

3. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Final simplification 99.3%

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

Alternative 5: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.102 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (or (<= x -0.102) (not (<= x 0.0011)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (/
      (+
       2.0
       (*
        (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
        (+ 1.0 (- (* -0.5 (* x x)) (cos y)))))
      (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((x <= -0.102) || !(x <= 0.0011)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    if ((x <= (-0.102d0)) .or. (.not. (x <= 0.0011d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + ((t_0 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))) * (1.0d0 + (((-0.5d0) * (x * x)) - cos(y))))) / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((x <= -0.102) || !(x <= 0.0011)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_0 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((t_0 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - Math.cos(y))))) / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	tmp = 0
	if (x <= -0.102) or not (x <= 0.0011):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_0 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + ((t_0 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - math.cos(y))))) / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if ((x <= -0.102) || !(x <= 0.0011))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_0 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	tmp = 0.0;
	if ((x <= -0.102) || ~((x <= 0.0011)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	else
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.102], N[Not[LessEqual[x, 0.0011]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.101999999999999993 or 0.00110000000000000007 < 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. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. add-sqr-sqrt 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in y around 0 60.7%

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

   if -0.101999999999999993 < x < 0.00110000000000000007

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. associate--l+ 99.7%

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

      2. unpow2 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.102 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 6: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-5} \lor \neg \left(x \leq 0.00062\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot t_0 + -0.0625 \cdot \left(x \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y))) (t_1 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -1.4e-5) (not (<= x 0.00062)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (+ (* (sin y) t_0) (* -0.0625 (* x t_0)))))
      (* 3.0 (+ 1.0 (- (+ t_1 (* (cos y) (- 1.5 t_1))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -1.4e-5) || !(x <= 0.00062)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - cos(y)
    t_1 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-1.4d-5)) .or. (.not. (x <= 0.00062d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * ((sin(y) * t_0) + ((-0.0625d0) * (x * t_0))))) / (3.0d0 * (1.0d0 + ((t_1 + (cos(y) * (1.5d0 - t_1))) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.cos(y);
	double t_1 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -1.4e-5) || !(x <= 0.00062)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * ((Math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_1 + (Math.cos(y) * (1.5 - t_1))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.cos(y)
	t_1 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -1.4e-5) or not (x <= 0.00062):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * ((math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_1 + (math.cos(y) * (1.5 - t_1))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -1.4e-5) || !(x <= 0.00062))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) * t_0) + Float64(-0.0625 * Float64(x * t_0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_1))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - cos(y);
	t_1 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -1.4e-5) || ~((x <= 0.00062)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -1.4e-5], N[Not[LessEqual[x, 0.00062]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999998e-5 or 6.2e-4 < x

   1. Initial program 98.9%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. add-sqr-sqrt 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in y around 0 61.0%

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

   if -1.39999999999999998e-5 < x < 6.2e-4

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-5} \lor \neg \left(x \leq 0.00062\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right) + -0.0625 \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 7: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.011 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_1 \cdot \left(\left(1 - \cos y\right) \cdot \left(x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t_0\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -0.011) (not (<= x 0.0011)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (* (sqrt 2.0) (sin x)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (/
      (fma
       (sqrt 2.0)
       (* t_1 (* (- 1.0 (cos y)) (+ x (* (sin y) -0.0625))))
       2.0)
      (+
       3.0
       (+
        (* 1.5 (* (cos x) t_0))
        (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -0.011) || !(x <= 0.0011)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_1 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), (t_1 * ((1.0 - cos(y)) * (x + (sin(y) * -0.0625)))), 2.0) / (3.0 + ((1.5 * (cos(x) * t_0)) + (9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0)))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -0.011) || !(x <= 0.0011))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_1 * Float64(Float64(1.0 - cos(y)) * Float64(x + Float64(sin(y) * -0.0625)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_0)) + Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.011], N[Not[LessEqual[x, 0.0011]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010999999999999999 or 0.00110000000000000007 < 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. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. add-sqr-sqrt 99.1%

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

      4. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in y around 0 60.7%

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

   if -0.010999999999999999 < x < 0.00110000000000000007

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.7%

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

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25} - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      3. div-inv 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \color{blue}{1.5}\right) \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. div-inv 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{1.5}\right)}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      7. div-inv 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot \color{blue}{1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot 1.5\right)}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      2. rem-square-sqrt 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{5} \cdot \left(1.5 \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - 5 \cdot \color{blue}{2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. cancel-sign-sub-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25 + \left(-5\right) \cdot 2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-11.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{9}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\sqrt{5} \cdot 1.5 + 4.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      9. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. Simplified 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   8. Taylor expanded in y around inf 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. add-log-exp 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\log \left(e^{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)}\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

      2. div-inv 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \log \left(e^{\left(\sin x - \color{blue}{\sin y \cdot \frac{1}{16}}\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \log \left(e^{\left(\sin x - \sin y \cdot \color{blue}{0.0625}\right) \cdot \left(\cos x - \cos y\right)}\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\log \left(e^{\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\cos x - \cos y\right)}\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

   11. Taylor expanded in x around 0 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sin y\right) + \left(1 - \cos y\right) \cdot x\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot -0.0625} + \left(1 - \cos y\right) \cdot x\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       2. associate-*l* 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right) \cdot \left(\sin y \cdot -0.0625\right)} + \left(1 - \cos y\right) \cdot x\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       3. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y \cdot \color{blue}{\left(-0.0625\right)}\right) + \left(1 - \cos y\right) \cdot x\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       4. distribute-rgt-neg-in 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\left(-\sin y \cdot 0.0625\right)} + \left(1 - \cos y\right) \cdot x\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       5. *-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(-\color{blue}{0.0625 \cdot \sin y}\right) + \left(1 - \cos y\right) \cdot x\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       6. distribute-lft-out 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\left(-0.0625 \cdot \sin y\right) + x\right)\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       7. distribute-lft-neg-in 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\color{blue}{\left(-0.0625\right) \cdot \sin y} + x\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

       8. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\color{blue}{-0.0625} \cdot \sin y + x\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

   13. Simplified 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sin y + x\right)\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.011 \lor \neg \left(x \leq 0.0011\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}\\ \end{array} \]

Alternative 8: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-5} \lor \neg \left(x \leq 7.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot t_0 + -0.0625 \cdot \left(x \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\left(t_2 + \cos y \cdot \left(1.5 - t_2\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -1.35e-5) (not (<= x 7.6e-5)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (+ (* (sin y) t_0) (* -0.0625 (* x t_0)))))
      (* 3.0 (+ 1.0 (- (+ t_2 (* (cos y) (- 1.5 t_2))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -1.35e-5) || !(x <= 7.6e-5)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (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 = 1.0d0 - cos(y)
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-1.35d-5)) .or. (.not. (x <= 7.6d-5))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(x) - cos(y))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * ((sin(y) * t_0) + ((-0.0625d0) * (x * t_0))))) / (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 = 1.0 - Math.cos(y);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -1.35e-5) || !(x <= 7.6e-5)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) - Math.cos(y))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * ((Math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_2 + (Math.cos(y) * (1.5 - t_2))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.cos(y)
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -1.35e-5) or not (x <= 7.6e-5):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) - math.cos(y))))) / (3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * ((math.sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (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(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -1.35e-5) || !(x <= 7.6e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) * t_0) + Float64(-0.0625 * Float64(x * t_0))))) / 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 = 1.0 - cos(y);
	t_1 = sqrt(5.0) / 2.0;
	t_2 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -1.35e-5) || ~((x <= 7.6e-5)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -1.35e-5], N[Not[LessEqual[x, 7.6e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(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 < -1.3499999999999999e-5 or 7.6000000000000004e-5 < x

   1. Initial program 98.9%

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

      1. 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+ 98.8%

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

      3. *-commutative 98.8%

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

      4. div-sub 98.8%

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

      5. metadata-eval 98.8%

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

      6. *-commutative 98.8%

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

      7. div-sub 98.8%

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

      8. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 60.9%

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

   if -1.3499999999999999e-5 < x < 7.6000000000000004e-5

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-5} \lor \neg \left(x \leq 7.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right) + -0.0625 \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 9: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \frac{\sqrt{5}}{2}\\ t_4 := \sin y - \frac{\sin x}{16}\\ t_5 := 1 - \cos y\\ \mathbf{if}\;x \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \cos y \cdot \left(1.5 - t_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.000235:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot t_5 + -0.0625 \cdot \left(x \cdot t_5\right)\right)}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_4 \cdot t_2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (* (sqrt 2.0) (sin x)))
        (t_3 (/ (sqrt 5.0) 2.0))
        (t_4 (- (sin y) (/ (sin x) 16.0)))
        (t_5 (- 1.0 (cos y))))
   (if (<= x -7e-6)
     (/
      (+ 2.0 (* t_2 (* t_4 t_0)))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (* (cos y) (- 1.5 t_3))))))
     (if (<= x 0.000235)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (+ (* (sin y) t_5) (* -0.0625 (* x t_5)))))
        (* 3.0 (+ 1.0 (- (+ t_1 (* (cos y) (- 1.5 t_1))) 0.5))))
       (/
        (+ 2.0 (* t_0 (* t_4 t_2)))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sqrt(2.0) * sin(x);
	double t_3 = sqrt(5.0) / 2.0;
	double t_4 = sin(y) - (sin(x) / 16.0);
	double t_5 = 1.0 - cos(y);
	double tmp;
	if (x <= -7e-6) {
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	} else if (x <= 0.000235) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_5) + (-0.0625 * (x * t_5))))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	} else {
		tmp = (2.0 + (t_0 * (t_4 * t_2))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = sqrt(5.0d0) * 0.5d0
    t_2 = sqrt(2.0d0) * sin(x)
    t_3 = sqrt(5.0d0) / 2.0d0
    t_4 = sin(y) - (sin(x) / 16.0d0)
    t_5 = 1.0d0 - cos(y)
    if (x <= (-7d-6)) then
        tmp = (2.0d0 + (t_2 * (t_4 * t_0))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_3 - 0.5d0)) + (cos(y) * (1.5d0 - t_3)))))
    else if (x <= 0.000235d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * ((sin(y) * t_5) + ((-0.0625d0) * (x * t_5))))) / (3.0d0 * (1.0d0 + ((t_1 + (cos(y) * (1.5d0 - t_1))) - 0.5d0)))
    else
        tmp = (2.0d0 + (t_0 * (t_4 * t_2))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sqrt(5.0) * 0.5;
	double t_2 = Math.sqrt(2.0) * Math.sin(x);
	double t_3 = Math.sqrt(5.0) / 2.0;
	double t_4 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_5 = 1.0 - Math.cos(y);
	double tmp;
	if (x <= -7e-6) {
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / (3.0 * (1.0 + ((Math.cos(x) * (t_3 - 0.5)) + (Math.cos(y) * (1.5 - t_3)))));
	} else if (x <= 0.000235) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * ((Math.sin(y) * t_5) + (-0.0625 * (x * t_5))))) / (3.0 * (1.0 + ((t_1 + (Math.cos(y) * (1.5 - t_1))) - 0.5)));
	} else {
		tmp = (2.0 + (t_0 * (t_4 * t_2))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sqrt(5.0) * 0.5
	t_2 = math.sqrt(2.0) * math.sin(x)
	t_3 = math.sqrt(5.0) / 2.0
	t_4 = math.sin(y) - (math.sin(x) / 16.0)
	t_5 = 1.0 - math.cos(y)
	tmp = 0
	if x <= -7e-6:
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / (3.0 * (1.0 + ((math.cos(x) * (t_3 - 0.5)) + (math.cos(y) * (1.5 - t_3)))))
	elif x <= 0.000235:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * ((math.sin(y) * t_5) + (-0.0625 * (x * t_5))))) / (3.0 * (1.0 + ((t_1 + (math.cos(y) * (1.5 - t_1))) - 0.5)))
	else:
		tmp = (2.0 + (t_0 * (t_4 * t_2))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sqrt(2.0) * sin(x))
	t_3 = Float64(sqrt(5.0) / 2.0)
	t_4 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_5 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (x <= -7e-6)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_4 * t_0))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_3))))));
	elseif (x <= 0.000235)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) * t_5) + Float64(-0.0625 * Float64(x * t_5))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_1))) - 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(t_4 * t_2))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sqrt(5.0) * 0.5;
	t_2 = sqrt(2.0) * sin(x);
	t_3 = sqrt(5.0) / 2.0;
	t_4 = sin(y) - (sin(x) / 16.0);
	t_5 = 1.0 - cos(y);
	tmp = 0.0;
	if (x <= -7e-6)
		tmp = (2.0 + (t_2 * (t_4 * t_0))) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	elseif (x <= 0.000235)
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_5) + (-0.0625 * (x * t_5))))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	else
		tmp = (2.0 + (t_0 * (t_4 * t_2))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e-6], N[(N[(2.0 + N[(t$95$2 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000235], 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[(N[(N[Sin[y], $MachinePrecision] * t$95$5), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.99999999999999989e-6

   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* 98.8%

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

      2. associate-+l+ 98.8%

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

      3. *-commutative 98.8%

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

      4. div-sub 98.8%

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

      5. metadata-eval 98.8%

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

      6. *-commutative 98.8%

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

      7. div-sub 98.8%

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

      8. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 67.7%

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

   if -6.99999999999999989e-6 < x < 2.34999999999999993e-4

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

   if 2.34999999999999993e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0 56.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.000235:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right) + -0.0625 \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 10: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-5} \lor \neg \left(x \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot t_0 + -0.0625 \cdot \left(x \cdot t_0\right)\right)}{3 \cdot \left(1 + \left(\left(t_1 + \cos y \cdot \left(1.5 - t_1\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y))) (t_1 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -1.35e-5) (not (<= x 8.5e-5)))
     (/
      (fma
       (sqrt 2.0)
       (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (+ (cos x) -1.0)))
       2.0)
      (+
       3.0
       (+
        (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0))))))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (+ (* (sin y) t_0) (* -0.0625 (* x t_0)))))
      (* 3.0 (+ 1.0 (- (+ t_1 (* (cos y) (- 1.5 t_1))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -1.35e-5) || !(x <= 8.5e-5)) {
		tmp = fma(sqrt(2.0), ((sin(y) - (sin(x) / 16.0)) * (sin(x) * (cos(x) + -1.0))), 2.0) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0)))))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_0) + (-0.0625 * (x * t_0))))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -1.35e-5) || !(x <= 8.5e-5))
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) * t_0) + Float64(-0.0625 * Float64(x * t_0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_1))) - 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -1.35e-5], N[Not[LessEqual[x, 8.5e-5]], $MachinePrecision]], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-5 or 8.500000000000001e-5 < x

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25} - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      3. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \color{blue}{1.5}\right) \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{1.5}\right)}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      7. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{0.6666666666666666}}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      8. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot \color{blue}{1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot 1.5\right)}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      2. rem-square-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{5} \cdot \left(1.5 \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      3. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - 5 \cdot \color{blue}{2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. cancel-sign-sub-inv 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25 + \left(-5\right) \cdot 2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-11.25}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{9}}{4.5 + \sqrt{5} \cdot 1.5}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      8. +-commutative 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\sqrt{5} \cdot 1.5 + 4.5}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      9. fma-def 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9}{\color{blue}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. Simplified 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{9}{\mathsf{fma}\left(\sqrt{5}, 1.5, 4.5\right)}}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   8. Taylor expanded in y around inf 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}} \]

   9. Taylor expanded in y around 0 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(\sin x \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 + \left(1.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)} \]

   if -1.3499999999999999e-5 < x < 8.500000000000001e-5

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-5} \lor \neg \left(x \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right) + -0.0625 \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 11: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{5} \cdot 0.5\\ t_3 := \cos x - \cos y\\ t_4 := 1 - \cos y\\ t_5 := \sqrt{2} \cdot -0.0625\\ t_6 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + t_5 \cdot \left(t_3 \cdot t_1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_6 + \cos y \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.000235:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot t_4 + -0.0625 \cdot \left(x \cdot t_4\right)\right)}{3 \cdot \left(1 + \left(\left(t_2 + \cos y \cdot \left(1.5 - t_2\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_5 \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_6}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 5.0) 0.5))
        (t_3 (- (cos x) (cos y)))
        (t_4 (- 1.0 (cos y)))
        (t_5 (* (sqrt 2.0) -0.0625))
        (t_6 (+ (sqrt 5.0) -1.0)))
   (if (<= x -1.4e-5)
     (/
      (* 0.3333333333333333 (+ 2.0 (* t_5 (* t_3 t_1))))
      (+ 1.0 (* 0.5 (+ (* (cos x) t_6) (* (cos y) t_0)))))
     (if (<= x 0.000235)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (+ (* (sin y) t_4) (* -0.0625 (* x t_4)))))
        (* 3.0 (+ 1.0 (- (+ t_2 (* (cos y) (- 1.5 t_2))) 0.5))))
       (/
        (+ 2.0 (* t_3 (* t_5 t_1)))
        (*
         3.0
         (+ (+ 1.0 (* (cos x) (/ t_6 2.0))) (* (cos y) (/ t_0 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(5.0) * 0.5;
	double t_3 = cos(x) - cos(y);
	double t_4 = 1.0 - cos(y);
	double t_5 = sqrt(2.0) * -0.0625;
	double t_6 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -1.4e-5) {
		tmp = (0.3333333333333333 * (2.0 + (t_5 * (t_3 * t_1)))) / (1.0 + (0.5 * ((cos(x) * t_6) + (cos(y) * t_0))));
	} else if (x <= 0.000235) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_4) + (-0.0625 * (x * t_4))))) / (3.0 * (1.0 + ((t_2 + (cos(y) * (1.5 - t_2))) - 0.5)));
	} else {
		tmp = (2.0 + (t_3 * (t_5 * t_1))) / (3.0 * ((1.0 + (cos(x) * (t_6 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sin(x) ** 2.0d0
    t_2 = sqrt(5.0d0) * 0.5d0
    t_3 = cos(x) - cos(y)
    t_4 = 1.0d0 - cos(y)
    t_5 = sqrt(2.0d0) * (-0.0625d0)
    t_6 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-1.4d-5)) then
        tmp = (0.3333333333333333d0 * (2.0d0 + (t_5 * (t_3 * t_1)))) / (1.0d0 + (0.5d0 * ((cos(x) * t_6) + (cos(y) * t_0))))
    else if (x <= 0.000235d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * ((sin(y) * t_4) + ((-0.0625d0) * (x * t_4))))) / (3.0d0 * (1.0d0 + ((t_2 + (cos(y) * (1.5d0 - t_2))) - 0.5d0)))
    else
        tmp = (2.0d0 + (t_3 * (t_5 * t_1))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_6 / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.pow(Math.sin(x), 2.0);
	double t_2 = Math.sqrt(5.0) * 0.5;
	double t_3 = Math.cos(x) - Math.cos(y);
	double t_4 = 1.0 - Math.cos(y);
	double t_5 = Math.sqrt(2.0) * -0.0625;
	double t_6 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -1.4e-5) {
		tmp = (0.3333333333333333 * (2.0 + (t_5 * (t_3 * t_1)))) / (1.0 + (0.5 * ((Math.cos(x) * t_6) + (Math.cos(y) * t_0))));
	} else if (x <= 0.000235) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * ((Math.sin(y) * t_4) + (-0.0625 * (x * t_4))))) / (3.0 * (1.0 + ((t_2 + (Math.cos(y) * (1.5 - t_2))) - 0.5)));
	} else {
		tmp = (2.0 + (t_3 * (t_5 * t_1))) / (3.0 * ((1.0 + (Math.cos(x) * (t_6 / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.pow(math.sin(x), 2.0)
	t_2 = math.sqrt(5.0) * 0.5
	t_3 = math.cos(x) - math.cos(y)
	t_4 = 1.0 - math.cos(y)
	t_5 = math.sqrt(2.0) * -0.0625
	t_6 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -1.4e-5:
		tmp = (0.3333333333333333 * (2.0 + (t_5 * (t_3 * t_1)))) / (1.0 + (0.5 * ((math.cos(x) * t_6) + (math.cos(y) * t_0))))
	elif x <= 0.000235:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * ((math.sin(y) * t_4) + (-0.0625 * (x * t_4))))) / (3.0 * (1.0 + ((t_2 + (math.cos(y) * (1.5 - t_2))) - 0.5)))
	else:
		tmp = (2.0 + (t_3 * (t_5 * t_1))) / (3.0 * ((1.0 + (math.cos(x) * (t_6 / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(5.0) * 0.5)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(1.0 - cos(y))
	t_5 = Float64(sqrt(2.0) * -0.0625)
	t_6 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -1.4e-5)
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(t_5 * Float64(t_3 * t_1)))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_6) + Float64(cos(y) * t_0)))));
	elseif (x <= 0.000235)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) * t_4) + Float64(-0.0625 * Float64(x * t_4))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_2 + Float64(cos(y) * Float64(1.5 - t_2))) - 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_3 * Float64(t_5 * t_1))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_6 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sin(x) ^ 2.0;
	t_2 = sqrt(5.0) * 0.5;
	t_3 = cos(x) - cos(y);
	t_4 = 1.0 - cos(y);
	t_5 = sqrt(2.0) * -0.0625;
	t_6 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -1.4e-5)
		tmp = (0.3333333333333333 * (2.0 + (t_5 * (t_3 * t_1)))) / (1.0 + (0.5 * ((cos(x) * t_6) + (cos(y) * t_0))));
	elseif (x <= 0.000235)
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * ((sin(y) * t_4) + (-0.0625 * (x * t_4))))) / (3.0 * (1.0 + ((t_2 + (cos(y) * (1.5 - t_2))) - 0.5)));
	else
		tmp = (2.0 + (t_3 * (t_5 * t_1))) / (3.0 * ((1.0 + (cos(x) * (t_6 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -1.4e-5], N[(N[(0.3333333333333333 * N[(2.0 + N[(t$95$5 * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$6), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000235], 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[(N[(N[Sin[y], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(-0.0625 * N[(x * t$95$4), $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], N[(N[(2.0 + N[(t$95$3 * N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$6 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999998e-5

   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. Taylor expanded in y around 0 65.3%

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

      1. associate-*r* 65.3%

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

      2. *-commutative 65.3%

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

   4. Simplified 65.3%

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

   5. Taylor expanded in x around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. associate-*r* 65.4%

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

      3. distribute-lft-out 65.4%

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

      4. sub-neg 65.4%

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

      5. metadata-eval 65.4%

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

      6. *-commutative 65.4%

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

   7. Simplified 65.4%

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

   if -1.39999999999999998e-5 < x < 2.34999999999999993e-4

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

   if 2.34999999999999993e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0 52.4%

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

      1. associate-*r* 52.4%

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

      2. *-commutative 52.4%

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

   4. Simplified 52.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.000235:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right) + -0.0625 \cdot \left(x \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 12: 78.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ t_2 := \sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\\ \mathbf{if}\;y \leq -0.00186 \lor \neg \left(y \leq 6.5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1
         (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
        (t_2 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))))
   (if (or (<= y -0.00186) (not (<= y 6.5e-33)))
     (/ (+ 2.0 (* t_2 (* (sin y) (- 1.0 (cos y))))) t_1)
     (/ (+ 2.0 (* t_2 (* (+ (cos x) -1.0) (+ y (* (sin x) -0.0625))))) t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0))));
	double t_2 = sqrt(2.0) * (sin(x) - (sin(y) / 16.0));
	double tmp;
	if ((y <= -0.00186) || !(y <= 6.5e-33)) {
		tmp = (2.0 + (t_2 * (sin(y) * (1.0 - cos(y))))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0))))
    t_2 = sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))
    if ((y <= (-0.00186d0)) .or. (.not. (y <= 6.5d-33))) then
        tmp = (2.0d0 + (t_2 * (sin(y) * (1.0d0 - cos(y))))) / t_1
    else
        tmp = (2.0d0 + (t_2 * ((cos(x) + (-1.0d0)) * (y + (sin(x) * (-0.0625d0)))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0))));
	double t_2 = Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0));
	double tmp;
	if ((y <= -0.00186) || !(y <= 6.5e-33)) {
		tmp = (2.0 + (t_2 * (Math.sin(y) * (1.0 - Math.cos(y))))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * ((Math.cos(x) + -1.0) * (y + (Math.sin(x) * -0.0625))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0))))
	t_2 = math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))
	tmp = 0
	if (y <= -0.00186) or not (y <= 6.5e-33):
		tmp = (2.0 + (t_2 * (math.sin(y) * (1.0 - math.cos(y))))) / t_1
	else:
		tmp = (2.0 + (t_2 * ((math.cos(x) + -1.0) * (y + (math.sin(x) * -0.0625))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0)))))
	t_2 = Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))
	tmp = 0.0
	if ((y <= -0.00186) || !(y <= 6.5e-33))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(sin(y) * Float64(1.0 - cos(y))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(cos(x) + -1.0) * Float64(y + Float64(sin(x) * -0.0625))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0))));
	t_2 = sqrt(2.0) * (sin(x) - (sin(y) / 16.0));
	tmp = 0.0;
	if ((y <= -0.00186) || ~((y <= 6.5e-33)))
		tmp = (2.0 + (t_2 * (sin(y) * (1.0 - cos(y))))) / t_1;
	else
		tmp = (2.0 + (t_2 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00186], N[Not[LessEqual[y, 6.5e-33]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0018600000000000001 or 6.4999999999999993e-33 < y

   1. Initial program 99.2%

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

      1. associate-*l* 99.1%

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

      2. associate-+l+ 99.1%

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

      3. *-commutative 99.1%

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

      4. div-sub 99.1%

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

      5. metadata-eval 99.1%

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

      6. *-commutative 99.1%

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

      7. div-sub 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 62.7%

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

   if -0.0018600000000000001 < y < 6.4999999999999993e-33

   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* 99.4%

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

      2. associate-+l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. div-sub 99.4%

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

      5. metadata-eval 99.4%

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

      6. *-commutative 99.4%

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

      7. div-sub 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-rgt-out 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00186 \lor \neg \left(y \leq 6.5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 13: 78.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 0.00017\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= y -0.0008) (not (<= y 0.00017)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (* (sin y) (- 1.0 (cos y)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (pow (sin x) 2.0) (+ (cos x) -1.0))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -0.0008) || !(y <= 0.00017)) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * (pow(sin(x), 2.0) * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((y <= (-0.0008d0)) .or. (.not. (y <= 0.00017d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) * (1.0d0 - cos(y))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -0.0008) || !(y <= 0.00017)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) * (1.0 - Math.cos(y))))) / (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) * -0.0625) * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (y <= -0.0008) or not (y <= 0.00017):
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) * (1.0 - math.cos(y))))) / (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) * -0.0625) * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.0008) || !(y <= 0.00017))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) * Float64(1.0 - cos(y))))) / 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) * -0.0625) * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((y <= -0.0008) || ~((y <= 0.00017)))
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) * (1.0 - cos(y))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((sin(x) ^ 2.0) * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0008], N[Not[LessEqual[y, 0.00017]], $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[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $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[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or 1.7e-4 < y

   1. Initial program 99.2%

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

      1. associate-*l* 99.1%

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

      2. associate-+l+ 99.1%

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

      3. *-commutative 99.1%

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

      4. div-sub 99.1%

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

      5. metadata-eval 99.1%

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

      6. *-commutative 99.1%

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

      7. div-sub 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 61.9%

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

   if -8.00000000000000038e-4 < y < 1.7e-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. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. add-sqr-sqrt 99.6%

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

      4. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 0.00017\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 14: 78.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \sqrt{2} \cdot -0.0625\\ t_3 := t_2 \cdot {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.00135:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot t_3}{t_1}\\ \mathbf{elif}\;y \leq 0.00017:\\ \;\;\;\;\frac{2 + t_2 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot t_3}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_1 (* 3.0 (+ t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (* (sqrt 2.0) -0.0625))
        (t_3 (* t_2 (pow (sin y) 2.0))))
   (if (<= y -0.00135)
     (/ (+ 2.0 (* (- 1.0 (cos y)) t_3)) t_1)
     (if (<= y 0.00017)
       (/
        (+ 2.0 (* t_2 (* (pow (sin x) 2.0) (+ (cos x) -1.0))))
        (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
       (/ (+ 2.0 (* (- (cos x) (cos y)) t_3)) t_1)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_1 = 3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = sqrt(2.0) * -0.0625;
	double t_3 = t_2 * pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.00135) {
		tmp = (2.0 + ((1.0 - cos(y)) * t_3)) / t_1;
	} else if (y <= 0.00017) {
		tmp = (2.0 + (t_2 * (pow(sin(x), 2.0) * (cos(x) + -1.0)))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * t_3)) / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_1 = 3.0d0 * (t_0 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = sqrt(2.0d0) * (-0.0625d0)
    t_3 = t_2 * (sin(y) ** 2.0d0)
    if (y <= (-0.00135d0)) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * t_3)) / t_1
    else if (y <= 0.00017d0) then
        tmp = (2.0d0 + (t_2 * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * t_3)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_1 = 3.0 * (t_0 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = Math.sqrt(2.0) * -0.0625;
	double t_3 = t_2 * Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.00135) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * t_3)) / t_1;
	} else if (y <= 0.00017) {
		tmp = (2.0 + (t_2 * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0)))) / (3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * t_3)) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_1 = 3.0 * (t_0 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = math.sqrt(2.0) * -0.0625
	t_3 = t_2 * math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.00135:
		tmp = (2.0 + ((1.0 - math.cos(y)) * t_3)) / t_1
	elif y <= 0.00017:
		tmp = (2.0 + (t_2 * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0)))) / (3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * t_3)) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_1 = Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(sqrt(2.0) * -0.0625)
	t_3 = Float64(t_2 * (sin(y) ^ 2.0))
	tmp = 0.0
	if (y <= -0.00135)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * t_3)) / t_1);
	elseif (y <= 0.00017)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * t_3)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_1 = 3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = sqrt(2.0) * -0.0625;
	t_3 = t_2 * (sin(y) ^ 2.0);
	tmp = 0.0;
	if (y <= -0.00135)
		tmp = (2.0 + ((1.0 - cos(y)) * t_3)) / t_1;
	elseif (y <= 0.00017)
		tmp = (2.0 + (t_2 * ((sin(x) ^ 2.0) * (cos(x) + -1.0)))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * t_3)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00135], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 0.00017], N[(N[(2.0 + N[(t$95$2 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0013500000000000001

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 61.2%

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

   3. Taylor expanded in x around 0 59.2%

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

      1. associate-*r* 59.2%

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

      2. *-commutative 59.2%

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

   5. Simplified 59.2%

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

   if -0.0013500000000000001 < y < 1.7e-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. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. add-sqr-sqrt 99.6%

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

      4. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   if 1.7e-4 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 63.7%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   4. Simplified 63.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00135:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;y \leq -0.00055 \lor \neg \left(y \leq 0.00017\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_0 \cdot {\sin y}^{2}\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (or (<= y -0.00055) (not (<= y 0.00017)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* t_0 (pow (sin y) 2.0))))
      (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+ 2.0 (* t_0 (* (pow (sin x) 2.0) (+ (cos x) -1.0))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((y <= -0.00055) || !(y <= 0.00017)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * pow(sin(y), 2.0)))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (t_0 * (pow(sin(x), 2.0) * (cos(x) + -1.0)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    if ((y <= (-0.00055d0)) .or. (.not. (y <= 0.00017d0))) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_0 * (sin(y) ** 2.0d0)))) / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + (t_0 * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if ((y <= -0.00055) || !(y <= 0.00017)) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_0 * Math.pow(Math.sin(y), 2.0)))) / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (t_0 * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0)))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	tmp = 0
	if (y <= -0.00055) or not (y <= 0.00017):
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_0 * math.pow(math.sin(y), 2.0)))) / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + (t_0 * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0)))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if ((y <= -0.00055) || !(y <= 0.00017))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_0 * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	tmp = 0.0;
	if ((y <= -0.00055) || ~((y <= 0.00017)))
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sin(y) ^ 2.0)))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + (t_0 * ((sin(x) ^ 2.0) * (cos(x) + -1.0)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00055], N[Not[LessEqual[y, 0.00017]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000033e-4 or 1.7e-4 < y

   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 x around 0 62.7%

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

   3. Taylor expanded in x around 0 61.3%

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

      1. associate-*r* 61.3%

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

      2. *-commutative 61.3%

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

   5. Simplified 61.3%

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

   if -5.50000000000000033e-4 < y < 1.7e-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. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. add-sqr-sqrt 99.6%

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

      4. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00055 \lor \neg \left(y \leq 0.00017\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 16: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.0005 \lor \neg \left(y \leq 6.5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_0 \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_0 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(t_1 + \cos x \cdot t_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -0.0005) (not (<= y 6.5e-33)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* t_0 (pow (sin y) 2.0))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_1 2.0)))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* t_0 (* (pow (sin x) 2.0) (+ (cos x) -1.0))))
       (+ 1.0 (* 0.5 (+ t_1 (* (cos x) t_2)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -0.0005) || !(y <= 6.5e-33)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (t_0 * (pow(sin(x), 2.0) * (cos(x) + -1.0)))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_2)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = sqrt(5.0d0) + (-1.0d0)
    if ((y <= (-0.0005d0)) .or. (.not. (y <= 6.5d-33))) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_0 * (sin(y) ** 2.0d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_2 / 2.0d0))) + (cos(y) * (t_1 / 2.0d0))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + (t_0 * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0))))) / (1.0d0 + (0.5d0 * (t_1 + (cos(x) * t_2)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -0.0005) || !(y <= 6.5e-33)) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_0 * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_2 / 2.0))) + (Math.cos(y) * (t_1 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (t_0 * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0)))) / (1.0 + (0.5 * (t_1 + (Math.cos(x) * t_2)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -0.0005) or not (y <= 6.5e-33):
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_0 * math.pow(math.sin(y), 2.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_2 / 2.0))) + (math.cos(y) * (t_1 / 2.0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (t_0 * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0)))) / (1.0 + (0.5 * (t_1 + (math.cos(x) * t_2)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -0.0005) || !(y <= 6.5e-33))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_0 * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(t_0 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))) / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(cos(x) * t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = 3.0 - sqrt(5.0);
	t_2 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -0.0005) || ~((y <= 6.5e-33)))
		tmp = (2.0 + ((1.0 - cos(y)) * (t_0 * (sin(y) ^ 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (t_0 * ((sin(x) ^ 2.0) * (cos(x) + -1.0)))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0005], N[Not[LessEqual[y, 6.5e-33]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(t$95$0 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000001e-4 or 6.4999999999999993e-33 < y

   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 x around 0 63.4%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. associate-*r* 62.1%

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

      2. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if -5.0000000000000001e-4 < y < 6.4999999999999993e-33

   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. Taylor expanded in y around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. *-commutative 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-out 98.6%

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

      6. sub-neg 98.6%

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

      7. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0005 \lor \neg \left(y \leq 6.5 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 17: 62.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 60.0%

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

   1. associate-*r* 60.0%

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

   2. *-commutative 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in y around 0 59.9%

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

6. Final simplification 59.9%

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

Alternative 18: 59.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + ((sqrt(2.0) * -0.0625) * (pow(sin(x), 2.0) * (cos(x) + -1.0)))) / (1.0 + (0.5 * ((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 = 0.3333333333333333d0 * ((2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0))))) / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 60.0%

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

   1. associate-*r* 60.0%

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

   2. *-commutative 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in y around 0 57.5%

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

   1. associate-*r* 57.5%

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

   2. *-commutative 57.5%

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

   3. sub-neg 57.5%

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

   4. metadata-eval 57.5%

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

   5. distribute-lft-out 57.5%

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

   6. sub-neg 57.5%

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

   7. metadata-eval 57.5%

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

7. Simplified 57.5%

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

8. Final simplification 57.5%

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

Alternative 19: 42.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 60.0%

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

   1. associate-*r* 60.0%

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

   2. *-commutative 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around 0 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)}} \]

   2. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}} \]

   3. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)} \]

   4. distribute-lft-in 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} + 0.5 \cdot -1\right)}} \]

   5. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right)} \]

   6. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{\left(-0.5\right)}\right)} \]

   7. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right)}} \]

   8. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

   9. +-commutative 42.0%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\left(\left(0.5 \cdot \sqrt{5} - 0.5\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   10. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{\left(0.5 \cdot \sqrt{5} + \left(-0.5\right)\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   11. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   12. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{0.5 \cdot -1}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   13. distribute-lft-in 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{0.5 \cdot \left(\sqrt{5} + -1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   14. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} + \color{blue}{\left(-1\right)}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   15. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \color{blue}{\left(\sqrt{5} - 1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

7. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 42.0%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \color{blue}{\left(\sqrt{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)} \cdot \sqrt{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)}\right)}} \]

   2. pow2 42.0%

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

   3. associate-+r+ 42.0%

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

   4. fma-def 42.0%

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

9. Applied egg-rr 42.0%

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

10. Final simplification 42.0%

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

Alternative 20: 42.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+ 1.0 (* 0.5 (+ (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.6666666666666666d0 / (1.0d0 + (0.5d0 * ((sqrt(5.0d0) + (-1.0d0)) + (cos(y) * (3.0d0 - sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + (0.5 * ((Math.sqrt(5.0) + -1.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return 0.6666666666666666 / (1.0 + (0.5 * ((math.sqrt(5.0) + -1.0) + (math.cos(y) * (3.0 - math.sqrt(5.0))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.6666666666666666 / (1.0 + (0.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (3.0 - sqrt(5.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 60.0%

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

   1. associate-*r* 60.0%

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

   2. *-commutative 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around 0 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)}} \]

   2. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}} \]

   3. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)} \]

   4. distribute-lft-in 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} + 0.5 \cdot -1\right)}} \]

   5. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right)} \]

   6. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{\left(-0.5\right)}\right)} \]

   7. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right)}} \]

   8. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

   9. +-commutative 42.0%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\left(\left(0.5 \cdot \sqrt{5} - 0.5\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   10. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{\left(0.5 \cdot \sqrt{5} + \left(-0.5\right)\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   11. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   12. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{0.5 \cdot -1}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   13. distribute-lft-in 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{0.5 \cdot \left(\sqrt{5} + -1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   14. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} + \color{blue}{\left(-1\right)}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   15. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \color{blue}{\left(\sqrt{5} - 1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

7. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)}} \]

8. Final simplification 42.0%

   \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

Alternative 21: 40.6% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 60.0%

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

   1. associate-*r* 60.0%

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

   2. *-commutative 60.0%

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

4. Simplified 60.0%

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

5. Taylor expanded in x around 0 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)}} \]

   2. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}} \]

   3. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + 0.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)} \]

   4. distribute-lft-in 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} + 0.5 \cdot -1\right)}} \]

   5. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right)} \]

   6. metadata-eval 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \left(0.5 \cdot \sqrt{5} + \color{blue}{\left(-0.5\right)}\right)} \]

   7. sub-neg 42.0%

      \[\leadsto \frac{0.6666666666666666}{\left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right) + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right)}} \]

   8. associate-+r+ 42.0%

      \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) + \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

   9. +-commutative 42.0%

      \[\leadsto \frac{0.6666666666666666}{1 + \color{blue}{\left(\left(0.5 \cdot \sqrt{5} - 0.5\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)}} \]

   10. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{\left(0.5 \cdot \sqrt{5} + \left(-0.5\right)\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   11. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{-0.5}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   12. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\left(0.5 \cdot \sqrt{5} + \color{blue}{0.5 \cdot -1}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   13. distribute-lft-in 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(\color{blue}{0.5 \cdot \left(\sqrt{5} + -1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   14. metadata-eval 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(\sqrt{5} + \color{blue}{\left(-1\right)}\right) + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

   15. sub-neg 42.0%

       \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \color{blue}{\left(\sqrt{5} - 1\right)} + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

7. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right)}} \]

8. Taylor expanded in y around 0 39.8%

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

9. Final simplification 39.8%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023202 
(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))))))