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

Percentage Accurate: 99.3% → 99.4%

Time: 42.1s

Alternatives: 23

Speedup: 0.9×

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, 0.7× speedup

Math

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

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
   (fma
    (cos y)
    (/ 9.0 (fma (sqrt 5.0) 1.5 4.5))
    (* (cos x) (/ 9.0 (fma 1.5 (sqrt 5.0) 1.5)))))))

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(sqrt(5.0), 1.5, 4.5)), (cos(x) * (9.0 / fma(1.5, sqrt(5.0), 1.5)))));
}

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around inf 99.4%

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

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. add-cbrt-cube 99.5%

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

   2. distribute-rgt-in 99.5%

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

   3. metadata-eval 99.5%

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

   4. distribute-rgt-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. add-cbrt-cube 99.4%

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

   2. flip-+ 99.2%

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

   3. metadata-eval 99.2%

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

10. Applied egg-rr 99.2%

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

    1. sub-neg 99.2%

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

    2. +-commutative 99.2%

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

    3. metadata-eval 99.2%

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

    4. swap-sqr 99.2%

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

    5. metadata-eval 99.2%

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

    6. *-commutative 99.2%

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

    7. rem-square-sqrt 99.5%

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

    8. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{11.25}}{\sqrt{5} \cdot 1.5 - -1.5}\right)} \]

    9. metadata-eval 99.5%

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

    10. *-commutative 99.5%

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

    11. fma-neg 99.5%

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

    12. metadata-eval 99.5%

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

12. Simplified 99.5%

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

    1. flip-- 99.5%

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

    2. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25} - \frac{\sqrt{5}}{0.6666666666666666} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    3. div-inv 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)} \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    4. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot \color{blue}{1.5}\right) \cdot \frac{\sqrt{5}}{0.6666666666666666}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    5. div-inv 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{0.6666666666666666}\right)}}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    6. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{1.5}\right)}{4.5 + \frac{\sqrt{5}}{0.6666666666666666}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    7. div-inv 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{0.6666666666666666}}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    8. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot \color{blue}{1.5}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

14. Applied egg-rr 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \color{blue}{\frac{20.25 - \left(\sqrt{5} \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]
15. Step-by-step derivation

    1. swap-sqr 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot 1.5\right)}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    2. rem-square-sqrt 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - \color{blue}{5} \cdot \left(1.5 \cdot 1.5\right)}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    3. metadata-eval 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 - 5 \cdot \color{blue}{2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    4. cancel-sign-sub-inv 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{20.25 + \left(-5\right) \cdot 2.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    5. metadata-eval 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-5} \cdot 2.25}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    6. metadata-eval 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{20.25 + \color{blue}{-11.25}}{4.5 + \sqrt{5} \cdot 1.5}, \cos x \cdot \frac{9}{\mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)}\right)} \]

    7. metadata-eval 99.6%

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

    8. +-commutative 99.6%

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

    9. fma-def 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

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

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

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around inf 99.4%

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

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. add-cbrt-cube 99.5%

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

   2. distribute-rgt-in 99.5%

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

   3. metadata-eval 99.5%

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

   4. distribute-rgt-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

   1. add-cbrt-cube 99.4%

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

   2. flip-+ 99.2%

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

   3. metadata-eval 99.2%

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

10. Applied egg-rr 99.2%

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

    1. sub-neg 99.2%

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

    2. +-commutative 99.2%

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

    3. metadata-eval 99.2%

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

    4. swap-sqr 99.2%

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

    5. metadata-eval 99.2%

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

    6. *-commutative 99.2%

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

    7. rem-square-sqrt 99.5%

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

    8. metadata-eval 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{11.25}}{\sqrt{5} \cdot 1.5 - -1.5}\right)} \]

    9. metadata-eval 99.5%

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

    10. *-commutative 99.5%

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

    11. fma-neg 99.5%

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

    12. metadata-eval 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 1.5\\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - t_0\right) + \cos x \cdot \left(t_0 + -1.5\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 1.5)))
   (/
    (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 t_0)) (* (cos x) (+ t_0 -1.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 1.5;
	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 - t_0)) + (cos(x) * (t_0 + -1.5))));
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 1.5)
	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 - t_0)) + Float64(cos(x) * Float64(t_0 + -1.5)))))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]}, N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(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 - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

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. Taylor expanded in x around inf 99.4%

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

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. distribute-rgt-in 99.4%

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

   5. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
   (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))
   2.0)
  (+
   3.0
   (+
    (* 6.0 (/ (cos y) (+ 3.0 (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(x) - (sin(y) / 16.0))), ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))), 2.0) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (cos(x) * (sqrt(5.0) + -1.0)))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
end

Wolfram

code[x_, y_] := 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[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. associate-+l+ 99.4%

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

   5. distribute-lft-in 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.3%

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

   1. flip-- 62.9%

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

   2. metadata-eval 62.9%

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

   3. add-sqr-sqrt 63.0%

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

   4. metadata-eval 63.0%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in y around inf 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(\cos x \cdot \left(\sqrt{5} \cdot 1.5 - 1.5\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))), ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))), 2.0) / (3.0 + ((cos(x) * ((sqrt(5.0) * 1.5) - 1.5)) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))), 2.0) / Float64(3.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 1.5) - 1.5)) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := 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[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. associate-+l+ 99.4%

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

   5. distribute-lft-in 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.3%

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

   1. flip-- 62.9%

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

   2. metadata-eval 62.9%

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

   3. add-sqr-sqrt 63.0%

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

   4. metadata-eval 63.0%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in y around inf 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. *-commutative 99.5%

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

   6. fma-def 99.5%

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

   7. sub-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. distribute-lft-in 99.5%

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

   10. metadata-eval 99.5%

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

   11. fma-def 99.5%

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

   12. associate-*r/ 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} \cdot 1.5\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.00042 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(t_3 \cdot t_2\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_3 \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot t_2\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - t_1\right) + 9 \cdot \frac{1}{1.5 + t_1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (* (sqrt 5.0) 1.5))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -0.00042) (not (<= x 8e-11)))
     (/
      (+ 2.0 (* (* t_3 t_2) (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (fma (sqrt 2.0) (* t_3 (* (- (sin x) (/ (sin y) 16.0)) t_2)) 2.0)
      (+ 3.0 (+ (* (cos y) (- 4.5 t_1)) (* 9.0 (/ 1.0 (+ 1.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sqrt(5.0) * 1.5;
	double t_2 = cos(x) - cos(y);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -0.00042) || !(x <= 8e-11)) {
		tmp = (2.0 + ((t_3 * t_2) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = fma(sqrt(2.0), (t_3 * ((sin(x) - (sin(y) / 16.0)) * t_2)), 2.0) / (3.0 + ((cos(y) * (4.5 - t_1)) + (9.0 * (1.0 / (1.5 + t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sqrt(5.0) * 1.5)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -0.00042) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(t_3 * t_2) * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_3 * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * t_2)), 2.0) / Float64(3.0 + Float64(Float64(cos(y) * Float64(4.5 - t_1)) + Float64(9.0 * Float64(1.0 / Float64(1.5 + t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $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[Or[LessEqual[x, -0.00042], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(t$95$3 * t$95$2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.5 - t$95$1), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(1.0 / N[(1.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000002e-4 or 7.99999999999999952e-11 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -4.2000000000000002e-4 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. add-cbrt-cube 99.8%

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

      2. distribute-rgt-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-rgt-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. add-cbrt-cube 99.6%

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

      2. flip-+ 99.6%

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

      3. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. sub-neg 99.6%

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

       2. +-commutative 99.6%

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

       3. metadata-eval 99.6%

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

       4. swap-sqr 99.6%

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

       5. metadata-eval 99.6%

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

       6. *-commutative 99.6%

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

       7. rem-square-sqrt 99.8%

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

       8. metadata-eval 99.8%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, 4.5 - \frac{\sqrt{5}}{0.6666666666666666}, \cos x \cdot \frac{-2.25 + \color{blue}{11.25}}{\sqrt{5} \cdot 1.5 - -1.5}\right)} \]

       9. metadata-eval 99.8%

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

       10. *-commutative 99.8%

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

       11. fma-neg 99.8%

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

       12. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 99.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00042 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(4.5 - \sqrt{5} \cdot 1.5\right) + 9 \cdot \frac{1}{1.5 + \sqrt{5} \cdot 1.5}\right)}\\ \end{array} \]

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     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 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	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 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    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 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	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.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	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.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	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 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-+l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

   7. div-sub 99.4%

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

   8. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 8: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-5} \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), t_1, 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))))
   (if (or (<= x -2.9e-5) (not (<= x 8e-11)))
     (/
      (+ 2.0 (* t_1 (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (fma (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) t_1 2.0)
      (+
       3.0
       (+
        (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
        (* 1.5 (+ (sqrt 5.0) -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = (sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y));
	double tmp;
	if ((x <= -2.9e-5) || !(x <= 8e-11)) {
		tmp = (2.0 + (t_1 * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = fma((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))), t_1, 2.0) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y)))
	tmp = 0.0
	if ((x <= -2.9e-5) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))), t_1, 2.0) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(sqrt(5.0) + -1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -2.9e-5], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e-5 or 7.99999999999999952e-11 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -2.9e-5 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. fma-def 99.7%

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

      4. associate-+l+ 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. metadata-eval 99.7%

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

   3. Simplified 99.6%

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

      1. flip-- 68.1%

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

      2. metadata-eval 68.1%

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

      3. add-sqr-sqrt 68.1%

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

      4. metadata-eval 68.1%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\sqrt{5} - 1\right) + 6 \cdot \frac{\cos y}{\sqrt{5} + 3}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-5} \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 9: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -8500000000 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(t_2 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot \left(1 - \cos y\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1
         (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -8500000000.0) (not (<= x 8e-11)))
     (/ (+ 2.0 (* (* t_2 (- (cos x) (cos y))) (* (sqrt 2.0) (sin x)))) t_1)
     (/
      (+
       2.0
       (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (* t_2 (- 1.0 (cos y)))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0))));
	double t_2 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -8500000000.0) || !(x <= 8e-11)) {
		tmp = (2.0 + ((t_2 * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / t_1;
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * (1.0 - cos(y))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0))))
    t_2 = sin(y) - (sin(x) / 16.0d0)
    if ((x <= (-8500000000.0d0)) .or. (.not. (x <= 8d-11))) then
        tmp = (2.0d0 + ((t_2 * (cos(x) - cos(y))) * (sqrt(2.0d0) * sin(x)))) / t_1
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_2 * (1.0d0 - cos(y))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0))));
	double t_2 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if ((x <= -8500000000.0) || !(x <= 8e-11)) {
		tmp = (2.0 + ((t_2 * (Math.cos(x) - Math.cos(y))) * (Math.sqrt(2.0) * Math.sin(x)))) / t_1;
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_2 * (1.0 - Math.cos(y))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0))))
	t_2 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if (x <= -8500000000.0) or not (x <= 8e-11):
		tmp = (2.0 + ((t_2 * (math.cos(x) - math.cos(y))) * (math.sqrt(2.0) * math.sin(x)))) / t_1
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_2 * (1.0 - math.cos(y))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0)))))
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -8500000000.0) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(cos(x) - cos(y))) * Float64(sqrt(2.0) * sin(x)))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_2 * Float64(1.0 - cos(y))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0))));
	t_2 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if ((x <= -8500000000.0) || ~((x <= 8e-11)))
		tmp = (2.0 + ((t_2 * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / t_1;
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * (1.0 - cos(y))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -8500000000.0], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5e9 or 7.99999999999999952e-11 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.8%

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

   if -8.5e9 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.7%

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

      3. *-commutative 99.7%

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

      4. div-sub 99.7%

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

      5. metadata-eval 99.7%

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

      6. *-commutative 99.7%

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

      7. div-sub 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8500000000 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0065 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0065) (not (<= x 8e-11)))
     (/
      (+
       2.0
       (*
        (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))
        (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- 1.0 (cos y)))
        (+ (* -0.0625 (pow (sin y) 2.0)) (* (sin y) (* x 1.00390625)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0065) || !(x <= 8e-11)) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (1.0 - cos(y))) * ((-0.0625 * pow(sin(y), 2.0)) + (sin(y) * (x * 1.00390625))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0065d0)) .or. (.not. (x <= 8d-11))) then
        tmp = (2.0d0 + (((sin(y) - (sin(x) / 16.0d0)) * (cos(x) - cos(y))) * (sqrt(2.0d0) * sin(x)))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (1.0d0 - cos(y))) * (((-0.0625d0) * (sin(y) ** 2.0d0)) + (sin(y) * (x * 1.00390625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0065) || !(x <= 8e-11)) {
		tmp = (2.0 + (((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) - Math.cos(y))) * (Math.sqrt(2.0) * Math.sin(x)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (1.0 - Math.cos(y))) * ((-0.0625 * Math.pow(Math.sin(y), 2.0)) + (Math.sin(y) * (x * 1.00390625))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0065) or not (x <= 8e-11):
		tmp = (2.0 + (((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) - math.cos(y))) * (math.sqrt(2.0) * math.sin(x)))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (1.0 - math.cos(y))) * ((-0.0625 * math.pow(math.sin(y), 2.0)) + (math.sin(y) * (x * 1.00390625))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0065) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))) * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(sin(y) * Float64(x * 1.00390625))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0065) || ~((x <= 8e-11)))
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (1.0 - cos(y))) * ((-0.0625 * (sin(y) ^ 2.0)) + (sin(y) * (x * 1.00390625))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0065], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[(x * 1.00390625), $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. Split input into 2 regimes

2. if x < -0.0064999999999999997 or 7.99999999999999952e-11 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.3%

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

   if -0.0064999999999999997 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-*r* 99.7%

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

      5. distribute-lft-out 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-lft1-in 99.7%

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

      8. associate-*r* 99.7%

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

      9. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0065 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 11: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -8500000000 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t_1\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (- (cos x) (cos y))))
   (if (or (<= x -8500000000.0) (not (<= x 8e-11)))
     (/
      (+ 2.0 (* (* (- (sin y) (/ (sin x) 16.0)) t_1) (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (* t_1 (* (+ (sin x) (* (sin y) -0.0625)) (* (sqrt 2.0) (sin y)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(x) - cos(y);
	double tmp;
	if ((x <= -8500000000.0) || !(x <= 8e-11)) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * t_1) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (t_1 * ((sin(x) + (sin(y) * -0.0625)) * (sqrt(2.0) * sin(y))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = cos(x) - cos(y)
    if ((x <= (-8500000000.0d0)) .or. (.not. (x <= 8d-11))) then
        tmp = (2.0d0 + (((sin(y) - (sin(x) / 16.0d0)) * t_1) * (sqrt(2.0d0) * sin(x)))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (t_1 * ((sin(x) + (sin(y) * (-0.0625d0))) * (sqrt(2.0d0) * sin(y))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -8500000000.0) || !(x <= 8e-11)) {
		tmp = (2.0 + (((Math.sin(y) - (Math.sin(x) / 16.0)) * t_1) * (Math.sqrt(2.0) * Math.sin(x)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (t_1 * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * (Math.sqrt(2.0) * Math.sin(y))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -8500000000.0) or not (x <= 8e-11):
		tmp = (2.0 + (((math.sin(y) - (math.sin(x) / 16.0)) * t_1) * (math.sqrt(2.0) * math.sin(x)))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (t_1 * ((math.sin(x) + (math.sin(y) * -0.0625)) * (math.sqrt(2.0) * math.sin(y))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -8500000000.0) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_1) * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(sqrt(2.0) * sin(y))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -8500000000.0) || ~((x <= 8e-11)))
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * t_1) * (sqrt(2.0) * sin(x)))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (t_1 * ((sin(x) + (sin(y) * -0.0625)) * (sqrt(2.0) * sin(y))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -8500000000.0], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $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. Split input into 2 regimes

2. if x < -8.5e9 or 7.99999999999999952e-11 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. associate-+l+ 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. div-sub 99.0%

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

      8. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 61.8%

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

   if -8.5e9 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 98.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8500000000 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \sin y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 12: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 1 + \cos x \cdot \frac{t_0}{2}\\ \mathbf{if}\;x \leq -0.0066:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot t_0\right) - \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (+ 1.0 (* (cos x) (/ t_0 2.0)))))
   (if (<= x -0.0066)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) (* (sqrt 2.0) -0.0625))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 8e-11)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- 1.0 (cos y)))
          (+ (* -0.0625 (pow (sin y) 2.0)) (* (sin y) (* x 1.00390625)))))
        (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
       (/
        (fma
         (sqrt 2.0)
         (*
          (- (sin y) (/ (sin x) 16.0))
          (* (- (sin x) (/ (sin y) 16.0)) (+ (cos x) -1.0)))
         2.0)
        (+ 3.0 (+ 4.5 (- (* (cos x) (* 1.5 t_0)) (* (sqrt 5.0) 1.5)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 1.0 + (cos(x) * (t_0 / 2.0));
	double tmp;
	if (x <= -0.0066) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + ((sqrt(2.0) * (1.0 - cos(y))) * ((-0.0625 * pow(sin(y), 2.0)) + (sin(y) * (x * 1.00390625))))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * (cos(x) + -1.0))), 2.0) / (3.0 + (4.5 + ((cos(x) * (1.5 * t_0)) - (sqrt(5.0) * 1.5))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0)))
	tmp = 0.0
	if (x <= -0.0066)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 8e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(sin(y) * Float64(x * 1.00390625))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = 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) + -1.0))), 2.0) / Float64(3.0 + Float64(4.5 + Float64(Float64(cos(x) * Float64(1.5 * t_0)) - Float64(sqrt(5.0) * 1.5)))));
	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[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0066], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-11], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[(x * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(4.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0066

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

   4. Simplified 57.4%

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

      1. flip-- 57.3%

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

      2. metadata-eval 57.3%

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

      3. add-sqr-sqrt 57.4%

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

      4. metadata-eval 57.4%

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

   6. Applied egg-rr 57.4%

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

   if -0.0066 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-*r* 99.7%

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

      5. distribute-lft-out 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-lft1-in 99.7%

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

      8. associate-*r* 99.7%

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

      9. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   if 7.99999999999999952e-11 < x

   1. Initial program 99.1%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 58.0%

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

      1. associate--l+ 57.9%

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

      2. associate-*r* 57.9%

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

      3. *-commutative 57.9%

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

      4. *-commutative 57.9%

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

      5. fma-neg 57.9%

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

      6. *-lft-identity 57.9%

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

      7. fma-neg 57.9%

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

      8. *-lft-identity 57.9%

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

      9. sub-neg 57.9%

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

      10. metadata-eval 57.9%

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

   6. Simplified 57.9%

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

   7. Taylor expanded in y around 0 58.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0066:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \]

Alternative 13: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.00275:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (pow (sin x) 2.0) (* (sqrt 2.0) -0.0625)))))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (<= x -0.00275)
     (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 8e-11)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- 1.0 (cos y)))
          (+ (* -0.0625 (pow (sin y) 2.0)) (* (sin y) (* x 1.00390625)))))
        t_2)
       (/ t_0 t_2)))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * (sqrt(2.0) * -0.0625)));
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if (x <= -0.00275) {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + ((sqrt(2.0) * (1.0 - cos(y))) * ((-0.0625 * pow(sin(y), 2.0)) + (sin(y) * (x * 1.00390625))))) / t_2;
	} else {
		tmp = t_0 / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (-0.0625d0))))
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = 3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if (x <= (-0.00275d0)) then
        tmp = t_0 / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 8d-11) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (1.0d0 - cos(y))) * (((-0.0625d0) * (sin(y) ** 2.0d0)) + (sin(y) * (x * 1.00390625d0))))) / t_2
    else
        tmp = t_0 / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * -0.0625)));
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if (x <= -0.00275) {
		tmp = t_0 / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (1.0 - Math.cos(y))) * ((-0.0625 * Math.pow(Math.sin(y), 2.0)) + (Math.sin(y) * (x * 1.00390625))))) / t_2;
	} else {
		tmp = t_0 / t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * -0.0625)))
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = 3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if x <= -0.00275:
		tmp = t_0 / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 8e-11:
		tmp = (2.0 + ((math.sqrt(2.0) * (1.0 - math.cos(y))) * ((-0.0625 * math.pow(math.sin(y), 2.0)) + (math.sin(y) * (x * 1.00390625))))) / t_2
	else:
		tmp = t_0 / t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * -0.0625))))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if (x <= -0.00275)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 8e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(sin(y) * Float64(x * 1.00390625))))) / t_2);
	else
		tmp = Float64(t_0 / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * (sqrt(2.0) * -0.0625)));
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = 3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if (x <= -0.00275)
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 8e-11)
		tmp = (2.0 + ((sqrt(2.0) * (1.0 - cos(y))) * ((-0.0625 * (sin(y) ^ 2.0)) + (sin(y) * (x * 1.00390625))))) / t_2;
	else
		tmp = t_0 / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00275], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-11], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * N[(x * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$0 / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0027499999999999998

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

   4. Simplified 57.4%

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

      1. flip-- 57.3%

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

      2. metadata-eval 57.3%

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

      3. add-sqr-sqrt 57.4%

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

      4. metadata-eval 57.4%

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

   6. Applied egg-rr 57.4%

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

   if -0.0027499999999999998 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-*r* 99.7%

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

      5. distribute-lft-out 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-lft1-in 99.7%

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

      8. associate-*r* 99.7%

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

      9. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   if 7.99999999999999952e-11 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   4. Simplified 57.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00275:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2} + \sin y \cdot \left(x \cdot 1.00390625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 14: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot t_0\right)\\ t_3 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_3 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right) \cdot \left(\sin y \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_3 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (+ 2.0 (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) t_0))))
        (t_3 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (<= x -0.00065)
     (/ t_2 (* 3.0 (+ t_3 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 8e-11)
       (/
        (+
         2.0
         (* (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y))) (* (sin y) t_0)))
        (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))
       (/ t_2 (* 3.0 (+ t_3 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = 2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * t_0));
	double t_3 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if (x <= -0.00065) {
		tmp = t_2 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))) * (sin(y) * t_0))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	} else {
		tmp = t_2 / (3.0 * (t_3 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = 2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * t_0))
    t_3 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    if (x <= (-0.00065d0)) then
        tmp = t_2 / (3.0d0 * (t_3 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 8d-11) then
        tmp = (2.0d0 + (((sin(y) - (sin(x) / 16.0d0)) * (1.0d0 - cos(y))) * (sin(y) * t_0))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    else
        tmp = t_2 / (3.0d0 * (t_3 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * t_0));
	double t_3 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if (x <= -0.00065) {
		tmp = t_2 / (3.0 * (t_3 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (((Math.sin(y) - (Math.sin(x) / 16.0)) * (1.0 - Math.cos(y))) * (Math.sin(y) * t_0))) / (3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	} else {
		tmp = t_2 / (3.0 * (t_3 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = 2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * t_0))
	t_3 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	tmp = 0
	if x <= -0.00065:
		tmp = t_2 / (3.0 * (t_3 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 8e-11:
		tmp = (2.0 + (((math.sin(y) - (math.sin(x) / 16.0)) * (1.0 - math.cos(y))) * (math.sin(y) * t_0))) / (3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	else:
		tmp = t_2 / (3.0 * (t_3 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * t_0)))
	t_3 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if (x <= -0.00065)
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 8e-11)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))) * Float64(sin(y) * t_0))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = sqrt(5.0) / 2.0;
	t_2 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * t_0));
	t_3 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	tmp = 0.0;
	if (x <= -0.00065)
		tmp = t_2 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 8e-11)
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))) * (sin(y) * t_0))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	else
		tmp = t_2 / (3.0 * (t_3 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00065], N[(t$95$2 / N[(3.0 * N[(t$95$3 + 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, 8e-11], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999997e-4

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

   4. Simplified 57.4%

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

      1. flip-- 57.3%

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

      2. metadata-eval 57.3%

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

      3. add-sqr-sqrt 57.4%

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

      4. metadata-eval 57.4%

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

   6. Applied egg-rr 57.4%

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

   if -6.4999999999999997e-4 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. associate-+l+ 99.7%

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

      3. *-commutative 99.7%

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

      4. div-sub 99.7%

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

      5. metadata-eval 99.7%

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

      6. *-commutative 99.7%

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

      7. div-sub 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

   if 7.99999999999999952e-11 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   4. Simplified 57.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.00122 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.00122) (not (<= x 8e-11)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) t_0))) t_1)
     (/ (+ 2.0 (* t_0 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))) t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00122) || !(x <= 8e-11)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.00122d0)) .or. (.not. (x <= 8d-11))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * t_0))) / t_1
    else
        tmp = (2.0d0 + (t_0 * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00122) || !(x <= 8e-11)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_0 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.00122) or not (x <= 8e-11):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * t_0))) / t_1
	else:
		tmp = (2.0 + (t_0 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.00122) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * t_0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.00122) || ~((x <= 8e-11)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * t_0))) / t_1;
	else
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00122], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00121999999999999995 or 7.99999999999999952e-11 < 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 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

   4. Simplified 57.5%

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

   if -0.00121999999999999995 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00122 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 16: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{2} \cdot -0.0625\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot t_1\right)\\ t_3 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_4 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + t_3\right)\\ \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_3 + \left(1 + \frac{\cos x}{\frac{2}{t_0}}\right)\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 2.0) -0.0625))
        (t_2 (+ 2.0 (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) t_1))))
        (t_3 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_4 (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) t_3))))
   (if (<= x -0.00105)
     (/ t_2 (* 3.0 (+ t_3 (+ 1.0 (/ (cos x) (/ 2.0 t_0))))))
     (if (<= x 8e-11)
       (/ (+ 2.0 (* t_1 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))) t_4)
       (/ t_2 t_4)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(2.0) * -0.0625;
	double t_2 = 2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * t_1));
	double t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_4 = 3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + t_3);
	double tmp;
	if (x <= -0.00105) {
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (cos(x) / (2.0 / t_0)))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (t_1 * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / t_4;
	} else {
		tmp = t_2 / t_4;
	}
	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 = sqrt(5.0d0) + (-1.0d0)
    t_1 = sqrt(2.0d0) * (-0.0625d0)
    t_2 = 2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * t_1))
    t_3 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    t_4 = 3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + t_3)
    if (x <= (-0.00105d0)) then
        tmp = t_2 / (3.0d0 * (t_3 + (1.0d0 + (cos(x) / (2.0d0 / t_0)))))
    else if (x <= 8d-11) then
        tmp = (2.0d0 + (t_1 * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / t_4
    else
        tmp = t_2 / t_4
    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(2.0) * -0.0625;
	double t_2 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * t_1));
	double t_3 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double t_4 = 3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + t_3);
	double tmp;
	if (x <= -0.00105) {
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (Math.cos(x) / (2.0 / t_0)))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (t_1 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / t_4;
	} else {
		tmp = t_2 / t_4;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = math.sqrt(2.0) * -0.0625
	t_2 = 2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * t_1))
	t_3 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	t_4 = 3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + t_3)
	tmp = 0
	if x <= -0.00105:
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (math.cos(x) / (2.0 / t_0)))))
	elif x <= 8e-11:
		tmp = (2.0 + (t_1 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / t_4
	else:
		tmp = t_2 / t_4
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(2.0) * -0.0625)
	t_2 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * t_1)))
	t_3 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_4 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + t_3))
	tmp = 0.0
	if (x <= -0.00105)
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_3 + Float64(1.0 + Float64(cos(x) / Float64(2.0 / t_0))))));
	elseif (x <= 8e-11)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_4);
	else
		tmp = Float64(t_2 / t_4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = sqrt(2.0) * -0.0625;
	t_2 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * t_1));
	t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	t_4 = 3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + t_3);
	tmp = 0.0;
	if (x <= -0.00105)
		tmp = t_2 / (3.0 * (t_3 + (1.0 + (cos(x) / (2.0 / t_0)))));
	elseif (x <= 8e-11)
		tmp = (2.0 + (t_1 * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_4;
	else
		tmp = t_2 / t_4;
	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[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00105], N[(t$95$2 / N[(3.0 * N[(t$95$3 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-11], N[(N[(2.0 + N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(t$95$2 / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00104999999999999994

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

   4. Simplified 57.4%

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

      1. associate-*l/ 57.4%

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

      2. *-commutative 57.4%

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

      3. sub-neg 57.4%

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

      4. metadata-eval 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. metadata-eval 57.4%

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

      2. sub-neg 57.4%

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

      3. associate-/l* 57.4%

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

      4. sub-neg 57.4%

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

      5. metadata-eval 57.4%

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

   8. Simplified 57.4%

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

   if -0.00104999999999999994 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

   7. Simplified 99.6%

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

   if 7.99999999999999952e-11 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   4. Simplified 57.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\cos x}{\frac{2}{\sqrt{5} + -1}}\right)\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 17: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot t_0\right)\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_3 := 3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.00075:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1 (+ 2.0 (* (- (cos x) (cos y)) (* (pow (sin x) 2.0) t_0))))
        (t_2 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_3 (* 3.0 (+ t_2 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (<= x -0.00075)
     (/ t_1 (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= x 8e-11)
       (/ (+ 2.0 (* t_0 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))) t_3)
       (/ t_1 t_3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = 2.0 + ((cos(x) - cos(y)) * (pow(sin(x), 2.0) * t_0));
	double t_2 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_3 = 3.0 * (t_2 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if (x <= -0.00075) {
		tmp = t_1 / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / t_3;
	} else {
		tmp = t_1 / 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(2.0d0) * (-0.0625d0)
    t_1 = 2.0d0 + ((cos(x) - cos(y)) * ((sin(x) ** 2.0d0) * t_0))
    t_2 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_3 = 3.0d0 * (t_2 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if (x <= (-0.00075d0)) then
        tmp = t_1 / (3.0d0 * (t_2 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (x <= 8d-11) then
        tmp = (2.0d0 + (t_0 * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / t_3
    else
        tmp = t_1 / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(x), 2.0) * t_0));
	double t_2 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_3 = 3.0 * (t_2 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if (x <= -0.00075) {
		tmp = t_1 / (3.0 * (t_2 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (x <= 8e-11) {
		tmp = (2.0 + (t_0 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / t_3;
	} else {
		tmp = t_1 / t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = 2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(x), 2.0) * t_0))
	t_2 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_3 = 3.0 * (t_2 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if x <= -0.00075:
		tmp = t_1 / (3.0 * (t_2 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif x <= 8e-11:
		tmp = (2.0 + (t_0 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / t_3
	else:
		tmp = t_1 / t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(x) ^ 2.0) * t_0)))
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_3 = Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if (x <= -0.00075)
		tmp = Float64(t_1 / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (x <= 8e-11)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_3);
	else
		tmp = Float64(t_1 / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) ^ 2.0) * t_0));
	t_2 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_3 = 3.0 * (t_2 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if (x <= -0.00075)
		tmp = t_1 / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (x <= 8e-11)
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_3;
	else
		tmp = t_1 / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = 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]}, If[LessEqual[x, -0.00075], N[(t$95$1 / 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, 8e-11], N[(N[(2.0 + N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$1 / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5000000000000002e-4

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

   4. Simplified 57.4%

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

      1. flip-- 57.3%

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

      2. metadata-eval 57.3%

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

      3. add-sqr-sqrt 57.4%

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

      4. metadata-eval 57.4%

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

   6. Applied egg-rr 57.4%

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

   if -7.5000000000000002e-4 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

   7. Simplified 99.6%

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

   if 7.99999999999999952e-11 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. *-commutative 57.6%

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

   4. Simplified 57.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00075:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 18: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-6} \lor \neg \left(y \leq 2.35 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot t_0\right) - \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -5.4e-6) (not (<= y 2.35e-24)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) (pow (sin y) 2.0))))
      (*
       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 (* (pow (sin x) 2.0) (+ (cos x) -1.0))) 2.0)
      (+ 3.0 (+ 4.5 (- (* (cos x) (* 1.5 t_0)) (* (sqrt 5.0) 1.5))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -5.4e-6) || !(y <= 2.35e-24)) {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * (pow(sin(x), 2.0) * (cos(x) + -1.0))), 2.0) / (3.0 + (4.5 + ((cos(x) * (1.5 * t_0)) - (sqrt(5.0) * 1.5))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -5.4e-6) || !(y <= 2.35e-24))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(4.5 + Float64(Float64(cos(x) * Float64(1.5 * t_0)) - Float64(sqrt(5.0) * 1.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -5.4e-6], N[Not[LessEqual[y, 2.35e-24]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(4.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.39999999999999997e-6 or 2.34999999999999996e-24 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r* 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. *-commutative 99.1%

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

      6. cancel-sign-sub-inv 99.1%

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

      7. metadata-eval 99.1%

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

      8. cancel-sign-sub-inv 99.1%

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

      9. metadata-eval 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. associate-*r* 59.3%

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

   7. Simplified 59.3%

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

   if -5.39999999999999997e-6 < y < 2.34999999999999996e-24

   1. Initial program 99.6%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. associate--l+ 99.5%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-neg 99.6%

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

      6. *-lft-identity 99.6%

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

      7. fma-neg 99.6%

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

      8. *-lft-identity 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 99.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-6} \lor \neg \left(y \leq 2.35 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5} \cdot 1.5\right)\right)}\\ \end{array} \]

Alternative 19: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.00105 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left({\sin x}^{2} \cdot t_0\right) \cdot \left(\cos x + -1\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.00105) (not (<= x 8e-11)))
     (/ (+ 2.0 (* (* (pow (sin x) 2.0) t_0) (+ (cos x) -1.0))) t_1)
     (/ (+ 2.0 (* t_0 (* (- 1.0 (cos y)) (pow (sin y) 2.0)))) t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00105) || !(x <= 8e-11)) {
		tmp = (2.0 + ((pow(sin(x), 2.0) * t_0) * (cos(x) + -1.0))) / t_1;
	} else {
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.00105d0)) .or. (.not. (x <= 8d-11))) then
        tmp = (2.0d0 + (((sin(x) ** 2.0d0) * t_0) * (cos(x) + (-1.0d0)))) / t_1
    else
        tmp = (2.0d0 + (t_0 * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.00105) || !(x <= 8e-11)) {
		tmp = (2.0 + ((Math.pow(Math.sin(x), 2.0) * t_0) * (Math.cos(x) + -1.0))) / t_1;
	} else {
		tmp = (2.0 + (t_0 * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.00105) or not (x <= 8e-11):
		tmp = (2.0 + ((math.pow(math.sin(x), 2.0) * t_0) * (math.cos(x) + -1.0))) / t_1
	else:
		tmp = (2.0 + (t_0 * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.00105) || !(x <= 8e-11))
		tmp = Float64(Float64(2.0 + Float64(Float64((sin(x) ^ 2.0) * t_0) * Float64(cos(x) + -1.0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.00105) || ~((x <= 8e-11)))
		tmp = (2.0 + (((sin(x) ^ 2.0) * t_0) * (cos(x) + -1.0))) / t_1;
	else
		tmp = (2.0 + (t_0 * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00105], N[Not[LessEqual[x, 8e-11]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00104999999999999994 or 7.99999999999999952e-11 < 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 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

   4. Simplified 57.5%

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

   5. Taylor expanded in y around 0 57.5%

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

   if -0.00104999999999999994 < x < 7.99999999999999952e-11

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. metadata-eval 99.7%

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

      8. cancel-sign-sub-inv 99.7%

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

      9. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00105 \lor \neg \left(x \leq 8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 20: 42.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5} \cdot 1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
  (+
   3.0
   (+ 4.5 (- (* (cos x) (* 1.5 (+ (sqrt 5.0) -1.0))) (* (sqrt 5.0) 1.5))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + (4.5 + ((cos(x) * (1.5 * (sqrt(5.0) + -1.0))) - (sqrt(5.0) * 1.5))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(4.5 + Float64(Float64(cos(x) * Float64(1.5 * Float64(sqrt(5.0) + -1.0))) - Float64(sqrt(5.0) * 1.5)))))
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

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. Taylor expanded in y around 0 61.1%

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

   1. associate--l+ 61.1%

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

   2. associate-*r* 61.1%

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

   3. *-commutative 61.1%

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

   4. *-commutative 61.1%

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

   5. fma-neg 61.1%

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

   6. *-lft-identity 61.1%

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

   7. fma-neg 61.1%

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

   8. *-lft-identity 61.1%

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

   9. sub-neg 61.1%

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

   10. metadata-eval 61.1%

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

6. Simplified 61.1%

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

7. Taylor expanded in x around 0 45.7%

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

8. Final simplification 45.7%

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

Alternative 21: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(4.5 + \left(\cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5} \cdot 1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (+ (cos x) -1.0))) 2.0)
  (+
   3.0
   (+ 4.5 (- (* (cos x) (* 1.5 (+ (sqrt 5.0) -1.0))) (* (sqrt 5.0) 1.5))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), (-0.0625 * (pow(sin(x), 2.0) * (cos(x) + -1.0))), 2.0) / (3.0 + (4.5 + ((cos(x) * (1.5 * (sqrt(5.0) + -1.0))) - (sqrt(5.0) * 1.5))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(4.5 + Float64(Float64(cos(x) * Float64(1.5 * Float64(sqrt(5.0) + -1.0))) - Float64(sqrt(5.0) * 1.5)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(4.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

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. Taylor expanded in y around 0 61.1%

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

   1. associate--l+ 61.1%

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

   2. associate-*r* 61.1%

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

   3. *-commutative 61.1%

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

   4. *-commutative 61.1%

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

   5. fma-neg 61.1%

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

   6. *-lft-identity 61.1%

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

   7. fma-neg 61.1%

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

   8. *-lft-identity 61.1%

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

   9. sub-neg 61.1%

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

   10. metadata-eval 61.1%

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

6. Simplified 61.1%

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

7. Taylor expanded in y around 0 60.7%

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

8. Final simplification 60.7%

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

Alternative 22: 35.8% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 63.0%

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

   1. associate-*r* 63.0%

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

   2. *-commutative 63.0%

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

4. Simplified 63.0%

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

5. Taylor expanded in x around 0 37.1%

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

   1. associate-*r* 37.1%

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

   2. *-commutative 37.1%

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

   3. unpow2 37.1%

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

7. Simplified 37.1%

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

8. Taylor expanded in x around 0 38.6%

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

9. Final simplification 38.6%

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

Alternative 23: 34.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 63.0%

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

   1. associate-*r* 63.0%

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

   2. *-commutative 63.0%

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

4. Simplified 63.0%

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

5. Taylor expanded in x around 0 37.1%

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

   1. associate-*r* 37.1%

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

   2. *-commutative 37.1%

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

   3. unpow2 37.1%

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

7. Simplified 37.1%

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

8. Taylor expanded in x around 0 37.1%

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

   1. sub-neg 37.1%

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

   2. metadata-eval 37.1%

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

   3. +-commutative 37.1%

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

   4. distribute-lft-in 37.1%

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

   5. metadata-eval 37.1%

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

10. Simplified 37.1%

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

11. Taylor expanded in x around 0 37.8%

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

12. Final simplification 37.8%

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

Reproduce

herbie shell --seed 2023258 
(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))))))