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

Percentage Accurate: 99.3% → 99.4%

Time: 42.4s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

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

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
   (fma
    (cos y)
    (/ 9.0 (fma 1.5 (sqrt 5.0) 4.5))
    (/ (* (cos x) (+ (sqrt 5.0) -1.0)) 0.6666666666666666)))))

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 + fma(cos(y), (9.0 / fma(1.5, sqrt(5.0), 4.5)), ((cos(x) * (sqrt(5.0) + -1.0)) / 0.6666666666666666)));
}

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 + fma(cos(y), Float64(9.0 / fma(1.5, sqrt(5.0), 4.5)), Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) / 0.6666666666666666))))
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[Cos[y], $MachinePrecision] * N[(9.0 / N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

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 99.3%

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

   3. metadata-eval 99.3%

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

   4. cancel-sign-sub-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

   8. +-commutative 99.3%

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

   9. *-commutative 99.3%

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

   10. fma-def 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

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
   (+
    (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0)))))
    (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))))))

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 + ((9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0))))) + (1.5 * (cos(x) * (sqrt(5.0) + -1.0)))));
}

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(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

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 99.3%

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

   3. metadata-eval 99.3%

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

   4. cancel-sign-sub-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

   8. +-commutative 99.3%

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

   9. *-commutative 99.3%

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

   10. fma-def 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in y around inf 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

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
   (+
    (* (cos y) (- 4.5 (* 1.5 (sqrt 5.0))))
    (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))))))

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 + ((cos(y) * (4.5 - (1.5 * sqrt(5.0)))) + (1.5 * (cos(x) * (sqrt(5.0) + -1.0)))));
}

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(cos(y) * Float64(4.5 - Float64(1.5 * sqrt(5.0)))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

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. fma-udef 99.3%

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

   2. div-inv 99.3%

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

   3. metadata-eval 99.3%

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

   4. div-inv 99.3%

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

   5. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

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 99.2%

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

   1. flip-- 42.2%

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

   2. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

3. Applied egg-rr 99.2%

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

   1. +-commutative 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. flip-- 42.2%

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

   2. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

3. Applied egg-rr 99.2%

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

   1. +-commutative 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

5. Simplified 99.2%

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

6. Taylor expanded in x around -inf 99.1%

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

7. Final simplification 99.1%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (*
     (- (cos x) (cos y))
     (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625))))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -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) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -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) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625d0)) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-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) * ((Math.cos(x) - Math.cos(y)) * ((Math.sin(x) - (Math.sin(y) * 0.0625)) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -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) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (math.sin(y) * 0.0625)) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -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(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(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) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(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 99.2%

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

2. Taylor expanded in x around -inf 99.2%

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

3. Final simplification 99.2%

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

Alternative 7: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

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 (- (cos x) (cos y)))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.078)
     (/
      (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (+
         2.0
         (*
          (* t_3 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
          (+ 1.0 (- (* -0.5 (* x x)) (cos y)))))
        t_1)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* t_2 (* (sin x) (- (sin y) (* (sin x) 0.0625))))))
        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 = cos(x) - cos(y);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.078) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / t_1;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(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) :: 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 = cos(x) - cos(y)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-0.078d0)) then
        tmp = (2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 2.65d-37) then
        tmp = (2.0d0 + ((t_3 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))) * (1.0d0 + (((-0.5d0) * (x * x)) - cos(y))))) / t_1
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * (sin(x) * (sin(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 = 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.cos(x) - Math.cos(y);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -0.078) {
		tmp = (2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - Math.cos(y))))) / t_1;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (Math.sin(x) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / 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.cos(x) - math.cos(y)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -0.078:
		tmp = (2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 2.65e-37:
		tmp = (2.0 + ((t_3 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - math.cos(y))))) / t_1
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (math.sin(x) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / 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(cos(x) - cos(y))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.078)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / 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 = cos(x) - cos(y);
	t_3 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -0.078)
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 2.65e-37)
		tmp = (2.0 + ((t_3 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 + ((-0.5 * (x * x)) - cos(y))))) / t_1;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(y) - (sin(x) * 0.0625)))))) / 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.078], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $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], If[LessEqual[x, 2.65e-37], N[(N[(2.0 + N[(N[(t$95$3 * 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] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0779999999999999999

   1. Initial program 99.0%

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

      1. flip-- 21.3%

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

      2. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   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. Step-by-step derivation

      1. +-commutative 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around 0 68.9%

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

   if -0.0779999999999999999 < x < 2.64999999999999998e-37

   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 x around 0 99.5%

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

      1. associate--l+ 99.5%

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

      2. unpow2 99.5%

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

   4. Simplified 99.5%

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

   if 2.64999999999999998e-37 < 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 67.0%

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

   3. Taylor expanded in x around inf 67.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.078:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0022:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right) + \sqrt{2} \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

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 (- (cos x) (cos y)))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0022)
     (/
      (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (+
         2.0
         (*
          (* t_3 (+ (* -0.0625 (* (sqrt 2.0) (sin y))) (* (sqrt 2.0) x)))
          (- 1.0 (cos y))))
        t_1)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* t_2 (* (sin x) (- (sin y) (* (sin x) 0.0625))))))
        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 = cos(x) - cos(y);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0022) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * ((-0.0625 * (sqrt(2.0) * sin(y))) + (sqrt(2.0) * x))) * (1.0 - cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(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) :: 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 = cos(x) - cos(y)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-0.0022d0)) then
        tmp = (2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 2.65d-37) then
        tmp = (2.0d0 + ((t_3 * (((-0.0625d0) * (sqrt(2.0d0) * sin(y))) + (sqrt(2.0d0) * x))) * (1.0d0 - cos(y)))) / t_1
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * (sin(x) * (sin(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 = 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.cos(x) - Math.cos(y);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -0.0022) {
		tmp = (2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * ((-0.0625 * (Math.sqrt(2.0) * Math.sin(y))) + (Math.sqrt(2.0) * x))) * (1.0 - Math.cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (Math.sin(x) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / 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.cos(x) - math.cos(y)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -0.0022:
		tmp = (2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 2.65e-37:
		tmp = (2.0 + ((t_3 * ((-0.0625 * (math.sqrt(2.0) * math.sin(y))) + (math.sqrt(2.0) * x))) * (1.0 - math.cos(y)))) / t_1
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (math.sin(x) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / 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(cos(x) - cos(y))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0022)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(Float64(-0.0625 * Float64(sqrt(2.0) * sin(y))) + Float64(sqrt(2.0) * x))) * Float64(1.0 - cos(y)))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / 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 = cos(x) - cos(y);
	t_3 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -0.0022)
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 2.65e-37)
		tmp = (2.0 + ((t_3 * ((-0.0625 * (sqrt(2.0) * sin(y))) + (sqrt(2.0) * x))) * (1.0 - cos(y)))) / t_1;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(y) - (sin(x) * 0.0625)))))) / 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0022], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $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], If[LessEqual[x, 2.65e-37], N[(N[(2.0 + N[(N[(t$95$3 * N[(N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00220000000000000013

   1. Initial program 99.0%

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

      1. flip-- 21.8%

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

      2. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   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. Step-by-step derivation

      1. +-commutative 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around 0 69.0%

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

   if -0.00220000000000000013 < x < 2.64999999999999998e-37

   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 x around 0 99.2%

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

   3. Taylor expanded in x around 0 99.2%

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

   if 2.64999999999999998e-37 < 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 67.0%

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

   3. Taylor expanded in x around inf 67.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0022:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \sin y\right) + \sqrt{2} \cdot x\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0031:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(t_3 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

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 (- (cos x) (cos y)))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0031)
     (/
      (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (+
         2.0
         (*
          (* t_3 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
          (- 1.0 (cos y))))
        t_1)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* t_2 (* (sin x) (- (sin y) (* (sin x) 0.0625))))))
        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 = cos(x) - cos(y);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0031) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(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) :: 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 = cos(x) - cos(y)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-0.0031d0)) then
        tmp = (2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 2.65d-37) then
        tmp = (2.0d0 + ((t_3 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))) * (1.0d0 - cos(y)))) / t_1
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * (sin(x) * (sin(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 = 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.cos(x) - Math.cos(y);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -0.0031) {
		tmp = (2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + ((t_3 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (1.0 - Math.cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (Math.sin(x) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / 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.cos(x) - math.cos(y)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -0.0031:
		tmp = (2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 2.65e-37:
		tmp = (2.0 + ((t_3 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))) * (1.0 - math.cos(y)))) / t_1
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (math.sin(x) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / 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(cos(x) - cos(y))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0031)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(1.0 - cos(y)))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / 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 = cos(x) - cos(y);
	t_3 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -0.0031)
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 2.65e-37)
		tmp = (2.0 + ((t_3 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (1.0 - cos(y)))) / t_1;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (sin(x) * (sin(y) - (sin(x) * 0.0625)))))) / 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0031], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $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], If[LessEqual[x, 2.65e-37], N[(N[(2.0 + N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00309999999999999989

   1. Initial program 99.0%

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

      1. flip-- 21.8%

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

      2. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   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. Step-by-step derivation

      1. +-commutative 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around 0 69.0%

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

   if -0.00309999999999999989 < x < 2.64999999999999998e-37

   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 x around 0 99.2%

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

   if 2.64999999999999998e-37 < 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 67.0%

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

   3. Taylor expanded in x around inf 67.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0031:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 10: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00098 \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.00098) (not (<= x 2.65e-37)))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (* (- (cos x) (cos y)) (* (sin x) (- (sin y) (* (sin x) 0.0625))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+
       3.0
       (+
        (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0)))))
        (* 1.5 (* (cos x) t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.00098) || !(x <= 2.65e-37)) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (sin(x) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0))))) + (1.5 * (cos(x) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.00098) || !(x <= 2.65e-37))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))) + Float64(1.5 * Float64(cos(x) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.00098], N[Not[LessEqual[x, 2.65e-37]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999997e-4 or 2.64999999999999998e-37 < 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 67.9%

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

   3. Taylor expanded in x around inf 68.0%

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

   if -9.7999999999999997e-4 < x < 2.64999999999999998e-37

   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

   3. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. div-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. swap-sqr 99.6%

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

      2. rem-square-sqrt 99.7%

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

      3. metadata-eval 99.7%

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

      4. cancel-sign-sub-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. fma-def 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 99.6%

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

   9. Taylor expanded in x around 0 99.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00098 \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 11: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} + -1\\ t_2 := 1 + \cos x \cdot \frac{t_1}{2}\\ \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (+ 1.0 (* (cos x) (/ t_1 2.0)))))
   (if (<= x -0.00088)
     (/
      (+ 2.0 (* t_0 (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+
         3.0
         (+
          (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0)))))
          (* 1.5 (* (cos x) t_1)))))
       (/
        (+
         2.0
         (* (sqrt 2.0) (* t_0 (* (sin x) (- (sin y) (* (sin x) 0.0625))))))
        (* 3.0 (+ t_2 (* (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) + -1.0;
	double t_2 = 1.0 + (cos(x) * (t_1 / 2.0));
	double tmp;
	if (x <= -0.00088) {
		tmp = (2.0 + (t_0 * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0))))) + (1.5 * (cos(x) * t_1))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * (sin(x) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * (t_2 + (cos(y) * ((3.0 - 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) + -1.0)
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0)))
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))) + Float64(1.5 * Float64(cos(x) * t_1)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return 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] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00088], N[(N[(2.0 + N[(t$95$0 * 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[(t$95$2 + 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], If[LessEqual[x, 2.65e-37], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + 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 < -8.80000000000000031e-4

   1. Initial program 99.0%

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

      1. flip-- 21.8%

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

      2. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   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. Step-by-step derivation

      1. +-commutative 21.8%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around 0 69.0%

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

   if -8.80000000000000031e-4 < x < 2.64999999999999998e-37

   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

   3. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. div-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. swap-sqr 99.6%

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

      2. rem-square-sqrt 99.7%

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

      3. metadata-eval 99.7%

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

      4. cancel-sign-sub-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. fma-def 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 99.6%

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

   9. Taylor expanded in x around 0 99.2%

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

   if 2.64999999999999998e-37 < 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 67.0%

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

   3. Taylor expanded in x around inf 67.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 12: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x \cdot t_0\\ t_2 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}{1 + 0.5 \cdot \left(t_1 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (cos x) t_0))
        (t_2 (* (+ (cos x) -1.0) (pow (sin x) 2.0))))
   (if (<= x -0.0032)
     (/
      (+ 2.0 (* (* (sqrt 2.0) -0.0625) t_2))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+ 3.0 (+ (* 9.0 (/ (cos y) (+ 4.5 (* 1.5 (sqrt 5.0))))) (* 1.5 t_1))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* (sqrt 2.0) t_2)))
         (+ 1.0 (* 0.5 (+ t_1 (* (cos y) (- 3.0 (sqrt 5.0))))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = cos(x) * t_0;
	double t_2 = (cos(x) + -1.0) * pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.0032) {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * t_2)) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((9.0 * (cos(y) / (4.5 + (1.5 * sqrt(5.0))))) + (1.5 * t_1)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * t_2))) / (1.0 + (0.5 * (t_1 + (cos(y) * (3.0 - sqrt(5.0)))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(cos(x) * t_0)
	t_2 = Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))
	tmp = 0.0
	if (x <= -0.0032)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * t_2)) / 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)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(9.0 * Float64(cos(y) / Float64(4.5 + Float64(1.5 * sqrt(5.0))))) + Float64(1.5 * t_1))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * t_2))) / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0032], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$2), $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], If[LessEqual[x, 2.65e-37], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(9.0 * N[(N[Cos[y], $MachinePrecision] / N[(4.5 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00320000000000000015

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. sub-neg 66.3%

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

      4. metadata-eval 66.3%

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

   4. Simplified 66.3%

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

      1. flip-- 21.3%

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

      2. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   6. Applied egg-rr 66.3%

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

      1. +-commutative 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   8. Simplified 66.3%

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

   if -0.00320000000000000015 < x < 2.64999999999999998e-37

   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

   3. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. div-inv 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. swap-sqr 99.6%

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

      2. rem-square-sqrt 99.6%

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

      3. metadata-eval 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 99.6%

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

   9. Taylor expanded in x around 0 98.9%

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

   if 2.64999999999999998e-37 < 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 63.5%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. sub-neg 63.5%

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

      4. metadata-eval 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. sub-neg 63.5%

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

      3. metadata-eval 63.5%

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

      4. distribute-lft-out 63.5%

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

      5. sub-neg 63.5%

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

      6. metadata-eval 63.5%

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

      7. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(9 \cdot \frac{\cos y}{4.5 + 1.5 \cdot \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 13: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;t_2 \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - t_1\right) + \cos x \cdot \left(t_1 - 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_3}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + 0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0032)
     (*
      t_2
      (/
       0.3333333333333333
       (+ 1.0 (+ (* (cos y) (- 1.5 t_1)) (* (cos x) (- t_1 0.5))))))
     (if (<= x 2.65e-37)
       (/
        (+
         2.0
         (* (* (- 1.0 (cos y)) (pow (sin y) 2.0)) (* (sqrt 2.0) -0.0625)))
        (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_3 2.0)))))
       (*
        0.3333333333333333
        (/ t_2 (+ 1.0 (* 0.5 (+ (* (cos x) t_0) (* (cos y) t_3))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0032) {
		tmp = t_2 * (0.3333333333333333 / (1.0 + ((cos(y) * (1.5 - t_1)) + (cos(x) * (t_1 - 0.5)))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + (((1.0 - cos(y)) * pow(sin(y), 2.0)) * (sqrt(2.0) * -0.0625))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_3 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * (t_2 / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * t_3)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = sqrt(5.0d0) * 0.5d0
    t_2 = 2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))
    t_3 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-0.0032d0)) then
        tmp = t_2 * (0.3333333333333333d0 / (1.0d0 + ((cos(y) * (1.5d0 - t_1)) + (cos(x) * (t_1 - 0.5d0)))))
    else if (x <= 2.65d-37) then
        tmp = (2.0d0 + (((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)) * (sqrt(2.0d0) * (-0.0625d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * (t_3 / 2.0d0))))
    else
        tmp = 0.3333333333333333d0 * (t_2 / (1.0d0 + (0.5d0 * ((cos(x) * t_0) + (cos(y) * t_3)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = Math.sqrt(5.0) * 0.5;
	double t_2 = 2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))));
	double t_3 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -0.0032) {
		tmp = t_2 * (0.3333333333333333 / (1.0 + ((Math.cos(y) * (1.5 - t_1)) + (Math.cos(x) * (t_1 - 0.5)))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + (((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)) * (Math.sqrt(2.0) * -0.0625))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * (t_3 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * (t_2 / (1.0 + (0.5 * ((Math.cos(x) * t_0) + (Math.cos(y) * t_3)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = math.sqrt(5.0) * 0.5
	t_2 = 2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))
	t_3 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -0.0032:
		tmp = t_2 * (0.3333333333333333 / (1.0 + ((math.cos(y) * (1.5 - t_1)) + (math.cos(x) * (t_1 - 0.5)))))
	elif x <= 2.65e-37:
		tmp = (2.0 + (((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)) * (math.sqrt(2.0) * -0.0625))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * (t_3 / 2.0))))
	else:
		tmp = 0.3333333333333333 * (t_2 / (1.0 + (0.5 * ((math.cos(x) * t_0) + (math.cos(y) * t_3)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0032)
		tmp = Float64(t_2 * Float64(0.3333333333333333 / Float64(1.0 + Float64(Float64(cos(y) * Float64(1.5 - t_1)) + Float64(cos(x) * Float64(t_1 - 0.5))))));
	elseif (x <= 2.65e-37)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * -0.0625))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_3 / 2.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_2 / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * t_3))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = sqrt(5.0) * 0.5;
	t_2 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))));
	t_3 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -0.0032)
		tmp = t_2 * (0.3333333333333333 / (1.0 + ((cos(y) * (1.5 - t_1)) + (cos(x) * (t_1 - 0.5)))));
	elseif (x <= 2.65e-37)
		tmp = (2.0 + (((1.0 - cos(y)) * (sin(y) ^ 2.0)) * (sqrt(2.0) * -0.0625))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_3 / 2.0))));
	else
		tmp = 0.3333333333333333 * (t_2 / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * t_3)))));
	end
	tmp_2 = 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[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0032], N[(t$95$2 * N[(0.3333333333333333 / N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-37], N[(N[(2.0 + N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $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[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$2 / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00320000000000000015

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. sub-neg 66.3%

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

      4. metadata-eval 66.3%

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

   4. Simplified 66.3%

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

      1. div-inv 66.2%

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

      2. associate-+l+ 66.2%

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

      3. *-commutative 66.2%

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

      4. div-sub 66.2%

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

      5. metadata-eval 66.2%

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

      6. *-commutative 66.2%

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

      7. div-sub 66.2%

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

      8. metadata-eval 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. associate-*l* 66.2%

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

      2. metadata-eval 66.2%

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

      3. sub-neg 66.2%

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

      4. *-commutative 66.2%

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

      5. *-commutative 66.2%

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

      6. sub-neg 66.2%

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

      7. metadata-eval 66.2%

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

      8. associate-/r* 66.3%

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

   8. Simplified 66.3%

      \[\leadsto \color{blue}{\left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}} \]

   9. Taylor expanded in x around inf 66.3%

      \[\leadsto \left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \color{blue}{\left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

   if -0.00320000000000000015 < x < 2.64999999999999998e-37

   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 x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   4. Simplified 98.8%

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

   if 2.64999999999999998e-37 < 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 63.5%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. sub-neg 63.5%

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

      4. metadata-eval 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. sub-neg 63.5%

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

      3. metadata-eval 63.5%

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

      4. distribute-lft-out 63.5%

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

      5. sub-neg 63.5%

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

      6. metadata-eval 63.5%

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

      7. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right) + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 14: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\cos x + -1\right) \cdot {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot -0.0625\\ t_3 := \sqrt{5} + -1\\ t_4 := 1 + \cos x \cdot \frac{t_3}{2}\\ \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + t_2 \cdot t_1}{3 \cdot \left(t_4 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot t_2}{3 \cdot \left(t_4 + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_3 + \cos y \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (+ (cos x) -1.0) (pow (sin x) 2.0)))
        (t_2 (* (sqrt 2.0) -0.0625))
        (t_3 (+ (sqrt 5.0) -1.0))
        (t_4 (+ 1.0 (* (cos x) (/ t_3 2.0)))))
   (if (<= x -0.0032)
     (/
      (+ 2.0 (* t_2 t_1))
      (* 3.0 (+ t_4 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 2.65e-37)
       (/
        (+ 2.0 (* (* (- 1.0 (cos y)) (pow (sin y) 2.0)) t_2))
        (* 3.0 (+ t_4 (* (cos y) (/ t_0 2.0)))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* (sqrt 2.0) t_1)))
         (+ 1.0 (* 0.5 (+ (* (cos x) t_3) (* (cos y) t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (cos(x) + -1.0) * pow(sin(x), 2.0);
	double t_2 = sqrt(2.0) * -0.0625;
	double t_3 = sqrt(5.0) + -1.0;
	double t_4 = 1.0 + (cos(x) * (t_3 / 2.0));
	double tmp;
	if (x <= -0.0032) {
		tmp = (2.0 + (t_2 * t_1)) / (3.0 * (t_4 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + (((1.0 - cos(y)) * pow(sin(y), 2.0)) * t_2)) / (3.0 * (t_4 + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * t_1))) / (1.0 + (0.5 * ((cos(x) * t_3) + (cos(y) * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = (cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0)
    t_2 = sqrt(2.0d0) * (-0.0625d0)
    t_3 = sqrt(5.0d0) + (-1.0d0)
    t_4 = 1.0d0 + (cos(x) * (t_3 / 2.0d0))
    if (x <= (-0.0032d0)) then
        tmp = (2.0d0 + (t_2 * t_1)) / (3.0d0 * (t_4 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 2.65d-37) then
        tmp = (2.0d0 + (((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)) * t_2)) / (3.0d0 * (t_4 + (cos(y) * (t_0 / 2.0d0))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * t_1))) / (1.0d0 + (0.5d0 * ((cos(x) * t_3) + (cos(y) * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = (Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0);
	double t_2 = Math.sqrt(2.0) * -0.0625;
	double t_3 = Math.sqrt(5.0) + -1.0;
	double t_4 = 1.0 + (Math.cos(x) * (t_3 / 2.0));
	double tmp;
	if (x <= -0.0032) {
		tmp = (2.0 + (t_2 * t_1)) / (3.0 * (t_4 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 2.65e-37) {
		tmp = (2.0 + (((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)) * t_2)) / (3.0 * (t_4 + (Math.cos(y) * (t_0 / 2.0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * t_1))) / (1.0 + (0.5 * ((Math.cos(x) * t_3) + (Math.cos(y) * t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = (math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0)
	t_2 = math.sqrt(2.0) * -0.0625
	t_3 = math.sqrt(5.0) + -1.0
	t_4 = 1.0 + (math.cos(x) * (t_3 / 2.0))
	tmp = 0
	if x <= -0.0032:
		tmp = (2.0 + (t_2 * t_1)) / (3.0 * (t_4 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 2.65e-37:
		tmp = (2.0 + (((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)) * t_2)) / (3.0 * (t_4 + (math.cos(y) * (t_0 / 2.0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * t_1))) / (1.0 + (0.5 * ((math.cos(x) * t_3) + (math.cos(y) * t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))
	t_2 = Float64(sqrt(2.0) * -0.0625)
	t_3 = Float64(sqrt(5.0) + -1.0)
	t_4 = Float64(1.0 + Float64(cos(x) * Float64(t_3 / 2.0)))
	tmp = 0.0
	if (x <= -0.0032)
		tmp = Float64(Float64(2.0 + Float64(t_2 * t_1)) / Float64(3.0 * Float64(t_4 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 2.65e-37)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)) * t_2)) / Float64(3.0 * Float64(t_4 + Float64(cos(y) * Float64(t_0 / 2.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * t_1))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_3) + Float64(cos(y) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = (cos(x) + -1.0) * (sin(x) ^ 2.0);
	t_2 = sqrt(2.0) * -0.0625;
	t_3 = sqrt(5.0) + -1.0;
	t_4 = 1.0 + (cos(x) * (t_3 / 2.0));
	tmp = 0.0;
	if (x <= -0.0032)
		tmp = (2.0 + (t_2 * t_1)) / (3.0 * (t_4 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 2.65e-37)
		tmp = (2.0 + (((1.0 - cos(y)) * (sin(y) ^ 2.0)) * t_2)) / (3.0 * (t_4 + (cos(y) * (t_0 / 2.0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * t_1))) / (1.0 + (0.5 * ((cos(x) * t_3) + (cos(y) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0032], N[(N[(2.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + 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], If[LessEqual[x, 2.65e-37], N[(N[(2.0 + N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00320000000000000015

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.3%

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

      1. associate-*r* 66.3%

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

      2. *-commutative 66.3%

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

      3. sub-neg 66.3%

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

      4. metadata-eval 66.3%

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

   4. Simplified 66.3%

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

      1. flip-- 21.3%

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

      2. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   6. Applied egg-rr 66.3%

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

      1. +-commutative 21.3%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   8. Simplified 66.3%

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

   if -0.00320000000000000015 < x < 2.64999999999999998e-37

   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 x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   4. Simplified 98.8%

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

   if 2.64999999999999998e-37 < 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 63.5%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. sub-neg 63.5%

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

      4. metadata-eval 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. sub-neg 63.5%

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

      3. metadata-eval 63.5%

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

      4. distribute-lft-out 63.5%

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

      5. sub-neg 63.5%

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

      6. metadata-eval 63.5%

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

      7. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 15: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-7} \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -4.1e-7) (not (<= x 2.65e-37)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
       (+ 1.0 (* 0.5 (+ (* (cos x) t_0) (* (cos y) (- 3.0 (sqrt 5.0))))))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
       (+ 1.0 (+ (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* t_0 0.5))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -4.1e-7) || !(x <= 2.65e-37)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 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) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-4.1d-7)) .or. (.not. (x <= 2.65d-37))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (1.0d0 + (0.5d0 * ((cos(x) * t_0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (t_0 * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -4.1e-7) || !(x <= 2.65e-37)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (1.0 + (0.5 * ((Math.cos(x) * t_0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (t_0 * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -4.1e-7) or not (x <= 2.65e-37):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (1.0 + (0.5 * ((math.cos(x) * t_0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (t_0 * 0.5))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -4.1e-7) || !(x <= 2.65e-37))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(t_0 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -4.1e-7) || ~((x <= 2.65e-37)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-7], N[Not[LessEqual[x, 2.65e-37]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0999999999999999e-7 or 2.64999999999999998e-37 < 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 65.1%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

      3. sub-neg 65.1%

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

      4. metadata-eval 65.1%

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

   4. Simplified 65.1%

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

   5. Taylor expanded in x around inf 65.1%

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

      1. *-commutative 65.1%

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

      2. sub-neg 65.1%

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

      3. metadata-eval 65.1%

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

      4. distribute-lft-out 65.1%

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

      5. sub-neg 65.1%

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

      6. metadata-eval 65.1%

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

      7. *-commutative 65.1%

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

   7. Simplified 65.1%

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

   if -4.0999999999999999e-7 < x < 2.64999999999999998e-37

   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-- 62.0%

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

      2. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. Applied egg-rr 99.5%

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

      1. +-commutative 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-7} \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 16: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - t_2\right) + \cos x \cdot \left(t_2 - 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_2 (* (sqrt 5.0) 0.5)))
   (if (<= x -8e-7)
     (*
      t_1
      (/
       0.3333333333333333
       (+ 1.0 (+ (* (cos y) (- 1.5 t_2)) (* (cos x) (- t_2 0.5))))))
     (if (<= x 2.65e-37)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
         (+ 1.0 (+ (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* t_0 0.5)))))
       (*
        0.3333333333333333
        (/
         t_1
         (+
          1.0
          (* 0.5 (+ (* (cos x) t_0) (* (cos y) (- 3.0 (sqrt 5.0))))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_2 = sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -8e-7) {
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((cos(y) * (1.5 - t_2)) + (cos(x) * (t_2 - 0.5)))));
	} else if (x <= 2.65e-37) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = 2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))
    t_2 = sqrt(5.0d0) * 0.5d0
    if (x <= (-8d-7)) then
        tmp = t_1 * (0.3333333333333333d0 / (1.0d0 + ((cos(y) * (1.5d0 - t_2)) + (cos(x) * (t_2 - 0.5d0)))))
    else if (x <= 2.65d-37) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (t_0 * 0.5d0))))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * ((cos(x) * t_0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = 2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))));
	double t_2 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -8e-7) {
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((Math.cos(y) * (1.5 - t_2)) + (Math.cos(x) * (t_2 - 0.5)))));
	} else if (x <= 2.65e-37) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (t_0 * 0.5))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((Math.cos(x) * t_0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))
	t_2 = math.sqrt(5.0) * 0.5
	tmp = 0
	if x <= -8e-7:
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((math.cos(y) * (1.5 - t_2)) + (math.cos(x) * (t_2 - 0.5)))))
	elif x <= 2.65e-37:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (t_0 * 0.5))))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((math.cos(x) * t_0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_2 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if (x <= -8e-7)
		tmp = Float64(t_1 * Float64(0.3333333333333333 / Float64(1.0 + Float64(Float64(cos(y) * Float64(1.5 - t_2)) + Float64(cos(x) * Float64(t_2 - 0.5))))));
	elseif (x <= 2.65e-37)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(t_0 * 0.5)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))));
	t_2 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if (x <= -8e-7)
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((cos(y) * (1.5 - t_2)) + (cos(x) * (t_2 - 0.5)))));
	elseif (x <= 2.65e-37)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -8e-7], N[(t$95$1 * N[(0.3333333333333333 / N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-37], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.9999999999999996e-7

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

      3. sub-neg 66.7%

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

      4. metadata-eval 66.7%

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

   4. Simplified 66.7%

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

      1. div-inv 66.7%

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

      2. associate-+l+ 66.7%

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

      3. *-commutative 66.7%

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

      4. div-sub 66.7%

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

      5. metadata-eval 66.7%

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

      6. *-commutative 66.7%

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

      7. div-sub 66.7%

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

      8. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*l* 66.7%

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

      2. metadata-eval 66.7%

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

      3. sub-neg 66.7%

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

      4. *-commutative 66.7%

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

      5. *-commutative 66.7%

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

      6. sub-neg 66.7%

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

      7. metadata-eval 66.7%

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

      8. associate-/r* 66.8%

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

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}} \]

   9. Taylor expanded in x around inf 66.8%

      \[\leadsto \left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \color{blue}{\left(\left(1.5 - 0.5 \cdot \sqrt{5}\right) \cdot \cos y + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}} \]

   if -7.9999999999999996e-7 < x < 2.64999999999999998e-37

   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-- 62.0%

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

      2. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. Applied egg-rr 99.5%

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

      1. +-commutative 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 2.64999999999999998e-37 < 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 63.5%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. sub-neg 63.5%

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

      4. metadata-eval 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in x around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. sub-neg 63.5%

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

      3. metadata-eval 63.5%

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

      4. distribute-lft-out 63.5%

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

      5. sub-neg 63.5%

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

      6. metadata-eval 63.5%

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

      7. *-commutative 63.5%

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

   7. Simplified 63.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\right) \cdot \frac{0.3333333333333333}{1 + \left(\cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right) + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 17: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-7} \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -8.5e-7) (not (<= x 2.65e-37)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0)))))
       (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) t_0))))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
       (+ 1.0 (+ (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* t_0 0.5))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -8.5e-7) || !(x <= 2.65e-37)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 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) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-8.5d-7)) .or. (.not. (x <= 2.65d-37))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * t_0)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (t_0 * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -8.5e-7) || !(x <= 2.65e-37)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * t_0)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (t_0 * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -8.5e-7) or not (x <= 2.65e-37):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * t_0)))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (t_0 * 0.5))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -8.5e-7) || !(x <= 2.65e-37))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * t_0))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(t_0 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -8.5e-7) || ~((x <= 2.65e-37)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -8.5e-7], N[Not[LessEqual[x, 2.65e-37]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000014e-7 or 2.64999999999999998e-37 < 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 65.1%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

      3. sub-neg 65.1%

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

      4. metadata-eval 65.1%

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

   4. Simplified 65.1%

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

   5. Taylor expanded in y around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. sub-neg 63.7%

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

      3. metadata-eval 63.7%

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

      4. distribute-lft-out 63.7%

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

      5. sub-neg 63.7%

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

      6. metadata-eval 63.7%

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

   7. Simplified 63.7%

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

   if -8.50000000000000014e-7 < x < 2.64999999999999998e-37

   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-- 62.0%

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

      2. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. Applied egg-rr 99.5%

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

      1. +-commutative 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-7} \lor \neg \left(x \leq 2.65 \cdot 10^{-37}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right) \cdot 0.5\right)}\\ \end{array} \]

Alternative 18: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \frac{0.3333333333333333}{1 + \left(\left(1.5 + \cos x \cdot \left(t_2 - 0.5\right)\right) - t_2\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0 \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (+ (cos x) -1.0) (pow (sin x) 2.0))))))
        (t_2 (* (sqrt 5.0) 0.5)))
   (if (<= x -7.2e-7)
     (*
      t_1
      (/ 0.3333333333333333 (+ 1.0 (- (+ 1.5 (* (cos x) (- t_2 0.5))) t_2))))
     (if (<= x 2.65e-37)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
         (+ 1.0 (+ (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* t_0 0.5)))))
       (*
        0.3333333333333333
        (/ t_1 (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))));
	double t_2 = sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -7.2e-7) {
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((1.5 + (cos(x) * (t_2 - 0.5))) - t_2)));
	} else if (x <= 2.65e-37) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = 2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))
    t_2 = sqrt(5.0d0) * 0.5d0
    if (x <= (-7.2d-7)) then
        tmp = t_1 * (0.3333333333333333d0 / (1.0d0 + ((1.5d0 + (cos(x) * (t_2 - 0.5d0))) - t_2)))
    else if (x <= 2.65d-37) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (t_0 * 0.5d0))))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = 2.0 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))));
	double t_2 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -7.2e-7) {
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((1.5 + (Math.cos(x) * (t_2 - 0.5))) - t_2)));
	} else if (x <= 2.65e-37) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (t_0 * 0.5))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 2.0 + (-0.0625 * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * math.pow(math.sin(x), 2.0))))
	t_2 = math.sqrt(5.0) * 0.5
	tmp = 0
	if x <= -7.2e-7:
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((1.5 + (math.cos(x) * (t_2 - 0.5))) - t_2)))
	elif x <= 2.65e-37:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (t_0 * 0.5))))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * (sin(x) ^ 2.0)))))
	t_2 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if (x <= -7.2e-7)
		tmp = Float64(t_1 * Float64(0.3333333333333333 / Float64(1.0 + Float64(Float64(1.5 + Float64(cos(x) * Float64(t_2 - 0.5))) - t_2))));
	elseif (x <= 2.65e-37)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(t_0 * 0.5)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) ^ 2.0))));
	t_2 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if (x <= -7.2e-7)
		tmp = t_1 * (0.3333333333333333 / (1.0 + ((1.5 + (cos(x) * (t_2 - 0.5))) - t_2)));
	elseif (x <= 2.65e-37)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((2.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (t_0 * 0.5))));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -7.2e-7], N[(t$95$1 * N[(0.3333333333333333 / N[(1.0 + N[(N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-37], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.19999999999999989e-7

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

      3. sub-neg 66.7%

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

      4. metadata-eval 66.7%

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

   4. Simplified 66.7%

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

      1. div-inv 66.7%

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

      2. associate-+l+ 66.7%

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

      3. *-commutative 66.7%

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

      4. div-sub 66.7%

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

      5. metadata-eval 66.7%

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

      6. *-commutative 66.7%

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

      7. div-sub 66.7%

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

      8. metadata-eval 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*l* 66.7%

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

      2. metadata-eval 66.7%

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

      3. sub-neg 66.7%

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

      4. *-commutative 66.7%

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

      5. *-commutative 66.7%

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

      6. sub-neg 66.7%

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

      7. metadata-eval 66.7%

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

      8. associate-/r* 66.8%

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

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \mathsf{fma}\left(\cos x, \frac{\sqrt{5}}{2} + -0.5, \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}} \]

   9. Taylor expanded in y around 0 65.8%

      \[\leadsto \left(2 + -0.0625 \cdot \left(\left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right)\right) \cdot \frac{0.3333333333333333}{1 + \color{blue}{\left(\left(1.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if -7.19999999999999989e-7 < x < 2.64999999999999998e-37

   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-- 62.0%

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

      2. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      3. add-sqr-sqrt 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

      4. metadata-eval 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. Applied egg-rr 99.5%

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

      1. +-commutative 62.0%

         \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 2.64999999999999998e-37 < 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 63.5%

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

      1. associate-*r* 63.5%

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

      2. *-commutative 63.5%

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

      3. sub-neg 63.5%

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

      4. metadata-eval 63.5%

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

   4. Simplified 63.5%

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

   5. Taylor expanded in y around 0 61.7%

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

      1. *-commutative 61.7%

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

      2. sub-neg 61.7%

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

      3. metadata-eval 61.7%

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

      4. distribute-lft-out 61.7%

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

      5. sub-neg 61.7%

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

      6. metadata-eval 61.7%

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

   7. Simplified 61.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;\left(2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\right) \cdot \frac{0.3333333333333333}{1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-37}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 19: 61.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((cos(x) + -1.0) * pow(sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 - 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 + ((-0.0625d0) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) ** 2.0d0))))) / (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 + (-0.0625 * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * Math.pow(Math.sin(x), 2.0))))) / (1.0 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))))));
}

Python

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[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 99.2%

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

2. Taylor expanded in y around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

   3. sub-neg 63.6%

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

   4. metadata-eval 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in y around 0 61.2%

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

   1. *-commutative 61.2%

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

   2. sub-neg 61.2%

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

   3. metadata-eval 61.2%

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

   4. distribute-lft-out 61.2%

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

   5. sub-neg 61.2%

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

   6. metadata-eval 61.2%

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

7. Simplified 61.2%

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

8. Final simplification 61.2%

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

Alternative 20: 42.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

   3. sub-neg 63.6%

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

   4. metadata-eval 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in x around 0 42.2%

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

   1. distribute-lft-out 42.2%

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

   2. *-commutative 42.2%

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

   3. sub-neg 42.2%

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

   4. metadata-eval 42.2%

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

7. Simplified 42.2%

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

   1. div-inv 42.2%

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

   2. *-commutative 42.2%

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

   3. fma-def 42.2%

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

9. Applied egg-rr 42.2%

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

10. Final simplification 42.2%

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

Alternative 21: 42.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+
   1.0
   (* 0.5 (+ (+ (sqrt 5.0) -1.0) (* (cos y) (/ 4.0 (+ 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) * (4.0 / (3.0 + sqrt(5.0)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.6666666666666666d0 / (1.0d0 + (0.5d0 * ((sqrt(5.0d0) + (-1.0d0)) + (cos(y) * (4.0d0 / (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) * (4.0 / (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) * (4.0 / (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(4.0 / 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) * (4.0 / (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[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

   3. sub-neg 63.6%

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

   4. metadata-eval 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in x around 0 42.2%

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

   1. distribute-lft-out 42.2%

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

   2. *-commutative 42.2%

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

   3. sub-neg 42.2%

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

   4. metadata-eval 42.2%

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

7. Simplified 42.2%

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

   1. flip-- 42.2%

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

   2. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   3. add-sqr-sqrt 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

   4. metadata-eval 42.2%

      \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}} + \left(\sqrt{5} + -1\right)\right)} \]

9. Applied egg-rr 42.2%

   \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}} + \left(\sqrt{5} + -1\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative 42.2%

       \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

11. Simplified 42.2%

    \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\cos y \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}} + \left(\sqrt{5} + -1\right)\right)} \]

12. Final simplification 42.2%

    \[\leadsto \frac{0.6666666666666666}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)} \]

Alternative 22: 42.9% 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 99.2%

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

2. Taylor expanded in y around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

   3. sub-neg 63.6%

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

   4. metadata-eval 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in x around 0 42.2%

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

   1. distribute-lft-out 42.2%

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

   2. *-commutative 42.2%

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

   3. sub-neg 42.2%

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

   4. metadata-eval 42.2%

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

7. Simplified 42.2%

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

8. Final simplification 42.2%

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

Alternative 23: 41.0% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. *-commutative 63.6%

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

   3. sub-neg 63.6%

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

   4. metadata-eval 63.6%

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

4. Simplified 63.6%

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

5. Taylor expanded in x around 0 42.2%

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

   1. distribute-lft-out 42.2%

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

   2. *-commutative 42.2%

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

   3. sub-neg 42.2%

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

   4. metadata-eval 42.2%

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

7. Simplified 42.2%

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

8. Taylor expanded in y around 0 40.4%

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

9. Final simplification 40.4%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023175 
(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))))))