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

Percentage Accurate: 99.3% → 99.3%

Time: 43.4s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

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

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)
  (fma
   (cos y)
   (/ 2.0 (+ 3.0 (sqrt 5.0)))
   (fma (cos x) (+ (/ (sqrt 5.0) 2.0) -0.5) 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) / fma(cos(y), (2.0 / (3.0 + sqrt(5.0))), fma(cos(x), ((sqrt(5.0) / 2.0) + -0.5), 1.0));
}

Julia

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

Wolfram

code[x_, y_] := N[(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] / 3.0), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $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-/r* 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. metadata-eval 99.3%

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

   2. div-sub 99.3%

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

   3. div-inv 99.3%

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

   4. flip-- 99.2%

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

   5. metadata-eval 99.2%

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

   6. associate-*l/ 99.2%

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

   7. sub-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. pow1/2 99.2%

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

   10. pow1/2 99.2%

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

   11. pow-sqr 99.4%

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

   12. metadata-eval 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

   16. metadata-eval 99.4%

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

   17. +-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \frac{4}{3 + \sqrt{5}}, 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma(sqrt(2.0), (((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))) * (sin(x) + (sin(y) * -0.0625))), 2.0) / (3.0 + fma(1.5, (cos(y) * (4.0 / (3.0 + sqrt(5.0)))), (6.0 * (cos(x) / (sqrt(5.0) + 1.0)))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) + Float64(sin(x) * -0.0625))) * Float64(sin(x) + Float64(sin(y) * -0.0625))), 2.0) / Float64(3.0 + fma(1.5, Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))), Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * 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%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. associate-/l* 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. div-inv 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. flip-- 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   6. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. associate-/r/ 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   9. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   10. sub-neg 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   11. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   12. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   13. pow-sqr 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   14. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   15. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   16. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   17. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   18. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. associate-/r* 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}} \]
11. Step-by-step derivation

    1. flip-- 63.7%

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

    2. sub-neg 63.7%

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

    3. metadata-eval 63.7%

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

    4. pow1/2 63.7%

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

    5. pow1/2 63.7%

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

    6. pow-sqr 63.8%

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

    7. metadata-eval 63.8%

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

    8. metadata-eval 63.8%

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

    9. metadata-eval 63.8%

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

    10. metadata-eval 63.8%

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

    11. +-commutative 63.8%

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

12. Applied egg-rr 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}, 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

13. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \frac{4}{3 + \sqrt{5}}, 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma(sqrt(2.0), (((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))) * (sin(x) + (sin(y) * -0.0625))), 2.0) / (3.0 + fma(1.5, (cos(y) * (3.0 - sqrt(5.0))), (6.0 * (cos(x) / (sqrt(5.0) + 1.0)))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) + Float64(sin(x) * -0.0625))) * Float64(sin(x) + Float64(sin(y) * -0.0625))), 2.0) / Float64(3.0 + fma(1.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * 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%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. associate-/l* 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. div-inv 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. flip-- 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   6. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. associate-/r/ 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   9. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   10. sub-neg 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   11. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   12. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   13. pow-sqr 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   14. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   15. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   16. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   17. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   18. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. associate-/r* 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot \sin x\right)\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))))) / (3.0d0 * (1.0d0 + ((cos(x) * ((sqrt(5.0d0) / 2.0d0) - 0.5d0)) + (cos(y) * (2.0d0 / (3.0d0 + sqrt(5.0d0)))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * (Math.sin(y) - (Math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) - 0.5)) + (Math.cos(y) * (2.0 / (3.0 + Math.sqrt(5.0)))))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * (math.sin(y) - (math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((math.cos(x) * ((math.sqrt(5.0) / 2.0) - 0.5)) + (math.cos(y) * (2.0 / (3.0 + math.sqrt(5.0)))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. metadata-eval 99.3%

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

   2. div-sub 99.3%

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

   3. div-inv 99.3%

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

   4. flip-- 99.2%

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

   5. metadata-eval 99.2%

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

   6. associate-*l/ 99.2%

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

   7. sub-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. pow1/2 99.2%

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

   10. pow1/2 99.2%

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

   11. pow-sqr 99.4%

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

   12. metadata-eval 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

   16. metadata-eval 99.4%

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

   17. +-commutative 99.4%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf 99.2%

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

   1. *-commutative 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. metadata-eval 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * (((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))) * (sin(x) + (sin(y) * -0.0625))))) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (6.0 * (cos(x) / (sqrt(5.0) + 1.0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * (((cos(x) - cos(y)) * (sin(y) + (sin(x) * (-0.0625d0)))) * (sin(x) + (sin(y) * (-0.0625d0)))))) / (3.0d0 + ((1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + (6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0)))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * (((Math.cos(x) - Math.cos(y)) * (Math.sin(y) + (Math.sin(x) * -0.0625))) * (Math.sin(x) + (Math.sin(y) * -0.0625))))) / (3.0 + ((1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + (6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0)))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * (((math.cos(x) - math.cos(y)) * (math.sin(y) + (math.sin(x) * -0.0625))) * (math.sin(x) + (math.sin(y) * -0.0625))))) / (3.0 + ((1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + (6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0)))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) + Float64(sin(x) * -0.0625))) * Float64(sin(x) + Float64(sin(y) * -0.0625))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * (((cos(x) - cos(y)) * (sin(y) + (sin(x) * -0.0625))) * (sin(x) + (sin(y) * -0.0625))))) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (6.0 * (cos(x) / (sqrt(5.0) + 1.0)))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * 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%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. associate-/l* 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. div-inv 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. flip-- 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   6. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. metadata-eval 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. associate-/r/ 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   9. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   10. sub-neg 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   11. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   12. pow1/2 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   13. pow-sqr 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   14. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   15. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   16. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   17. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   18. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. associate-/r* 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   2. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   3. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

9. Final simplification 99.3%

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{\cos x}{\sqrt{5} + 1}\right)} \]

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (sqrt 2.0)
     (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))))
  (*
   3.0
   (+
    1.0
    (+
     (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
     (* (cos y) (- 1.5 (sqrt 1.25))))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (1.5 - sqrt(1.25))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))))) / (3.0d0 * (1.0d0 + ((cos(x) * ((sqrt(5.0d0) / 2.0d0) - 0.5d0)) + (cos(y) * (1.5d0 - sqrt(1.25d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * (Math.sin(y) - (Math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) - 0.5)) + (Math.cos(y) * (1.5 - Math.sqrt(1.25))))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * (math.sin(y) - (math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((math.cos(x) * ((math.sqrt(5.0) / 2.0) - 0.5)) + (math.cos(y) * (1.5 - math.sqrt(1.25))))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(1.5 - sqrt(1.25)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (1.5 - sqrt(1.25))))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. sub-neg 99.3%

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

   2. add-sqr-sqrt 99.2%

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

   3. sqrt-unprod 99.3%

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

   4. frac-times 99.3%

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

   5. pow1/2 99.3%

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

   6. pow1/2 99.3%

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

   7. pow-sqr 99.3%

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

   8. metadata-eval 99.3%

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

   9. metadata-eval 99.3%

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

   10. metadata-eval 99.3%

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 + \left(-\sqrt{\frac{5}{\color{blue}{4}}}\right)\right)\right)\right)} \]

   11. metadata-eval 99.3%

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 + \left(-\sqrt{\color{blue}{1.25}}\right)\right)\right)\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\left(1.5 + \left(-\sqrt{1.25}\right)\right)}\right)\right)} \]
6. Step-by-step derivation

   1. sub-neg 99.3%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\left(1.5 - \sqrt{1.25}\right)}\right)\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\left(1.5 - \sqrt{1.25}\right)}\right)\right)} \]

8. Final simplification 99.3%

   \[\leadsto \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{1.25}\right)\right)\right)} \]

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 2 + \left(\cos x - \cos y\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.0026:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot t_2\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{t_0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{t_0}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (+ 2.0 (* (- (cos x) (cos y)) (* t_2 (* (sqrt 2.0) (sin x)))))))
   (if (<= x -0.0026)
     (/ t_3 (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 1.1e-6)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* (- (sin x) (/ (sin y) 16.0)) t_2))
          (- 1.0 (cos y))))
        (*
         3.0
         (+
          1.0
          (+ (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5)) (* (cos y) (/ 2.0 t_0))))))
       (/ t_3 (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 t_0) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = sin(y) - (sin(x) / 16.0);
	double t_3 = 2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * sin(x))));
	double tmp;
	if (x <= -0.0026) {
		tmp = t_3 / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 1.1e-6) {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * t_2)) * (1.0 - cos(y)))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / t_0)))));
	} else {
		tmp = t_3 / (3.0 * (t_1 + (cos(y) * ((4.0 / t_0) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = sin(y) - (sin(x) / 16.0d0)
    t_3 = 2.0d0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0d0) * sin(x))))
    if (x <= (-0.0026d0)) then
        tmp = t_3 / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (x <= 1.1d-6) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * t_2)) * (1.0d0 - cos(y)))) / (3.0d0 * (1.0d0 + ((cos(x) * ((sqrt(5.0d0) / 2.0d0) - 0.5d0)) + (cos(y) * (2.0d0 / t_0)))))
    else
        tmp = t_3 / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / t_0) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_3 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (t_2 * (Math.sqrt(2.0) * Math.sin(x))));
	double tmp;
	if (x <= -0.0026) {
		tmp = t_3 / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (x <= 1.1e-6) {
		tmp = (2.0 + ((Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * t_2)) * (1.0 - Math.cos(y)))) / (3.0 * (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) - 0.5)) + (Math.cos(y) * (2.0 / t_0)))));
	} else {
		tmp = t_3 / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / t_0) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = math.sin(y) - (math.sin(x) / 16.0)
	t_3 = 2.0 + ((math.cos(x) - math.cos(y)) * (t_2 * (math.sqrt(2.0) * math.sin(x))))
	tmp = 0
	if x <= -0.0026:
		tmp = t_3 / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif x <= 1.1e-6:
		tmp = (2.0 + ((math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * t_2)) * (1.0 - math.cos(y)))) / (3.0 * (1.0 + ((math.cos(x) * ((math.sqrt(5.0) / 2.0) - 0.5)) + (math.cos(y) * (2.0 / t_0)))))
	else:
		tmp = t_3 / (3.0 * (t_1 + (math.cos(y) * ((4.0 / t_0) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_2 * Float64(sqrt(2.0) * sin(x)))))
	tmp = 0.0
	if (x <= -0.0026)
		tmp = Float64(t_3 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 1.1e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * t_2)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(2.0 / t_0))))));
	else
		tmp = Float64(t_3 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / t_0) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = sin(y) - (sin(x) / 16.0);
	t_3 = 2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * sin(x))));
	tmp = 0.0;
	if (x <= -0.0026)
		tmp = t_3 / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (x <= 1.1e-6)
		tmp = (2.0 + ((sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * t_2)) * (1.0 - cos(y)))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / t_0)))));
	else
		tmp = t_3 / (3.0 * (t_1 + (cos(y) * ((4.0 / t_0) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0026], N[(t$95$3 / 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], If[LessEqual[x, 1.1e-6], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0025999999999999999

   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 62.2%

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

      1. *-commutative 62.2%

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

   4. Simplified 62.2%

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

   if -0.0025999999999999999 < x < 1.1000000000000001e-6

   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 99.6%

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

      1. metadata-eval 99.7%

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

      2. div-sub 99.7%

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

      3. div-inv 99.7%

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

      4. flip-- 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. pow1/2 99.6%

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

      10. pow1/2 99.6%

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

      11. pow-sqr 99.7%

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

      12. metadata-eval 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 1.1000000000000001e-6 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.2%

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

      1. *-commutative 63.2%

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

   4. Simplified 63.2%

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

      1. flip-- 63.1%

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

      2. sub-neg 63.1%

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

      3. metadata-eval 63.1%

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

      4. pow1/2 63.1%

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

      5. pow1/2 63.1%

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

      6. pow-sqr 63.2%

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

      7. metadata-eval 63.2%

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

      8. metadata-eval 63.2%

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

      9. metadata-eval 63.2%

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

      10. metadata-eval 63.2%

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

      11. +-commutative 63.2%

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

   6. Applied egg-rr 63.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0026:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.000185:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x))))))
        (t_2 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (<= x -0.000185)
     (/ t_1 (* 3.0 (+ t_2 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 1.1e-6)
       (/
        (+
         2.0
         (*
          (- 1.0 (cos y))
          (*
           (sqrt 2.0)
           (fma -0.0625 (pow (sin y) 2.0) (* x (* (sin y) 1.00390625))))))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (/
        t_1
        (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))));
	double t_2 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if (x <= -0.000185) {
		tmp = t_1 / (3.0 * (t_2 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 1.1e-6) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) * 1.00390625)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = t_1 / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x)))))
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if (x <= -0.000185)
		tmp = Float64(t_1 / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 1.1e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		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)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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]}, If[LessEqual[x, -0.000185], N[(t$95$1 / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-6], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.85e-4

   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 62.2%

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

      1. *-commutative 62.2%

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

   4. Simplified 62.2%

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

   if -1.85e-4 < x < 1.1000000000000001e-6

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
   7. Step-by-step derivation

      1. fma-def 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      2. distribute-rgt1-in 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\sin y \cdot \left(0.00390625 + 1\right)\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot \color{blue}{1.00390625}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

   8. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

   if 1.1000000000000001e-6 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.2%

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

      1. *-commutative 63.2%

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

   4. Simplified 63.2%

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

      1. flip-- 63.1%

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

      2. sub-neg 63.1%

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

      3. metadata-eval 63.1%

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

      4. pow1/2 63.1%

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

      5. pow1/2 63.1%

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

      6. pow-sqr 63.2%

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

      7. metadata-eval 63.2%

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

      8. metadata-eval 63.2%

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

      9. metadata-eval 63.2%

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

      10. metadata-eval 63.2%

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

      11. +-commutative 63.2%

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

   6. Applied egg-rr 63.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000185:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -2.5e-5) (not (<= x 1.1e-6)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (fma -0.0625 (pow (sin y) 2.0) (* x (* (sin y) 1.00390625))))))
      (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -2.5e-5) || !(x <= 1.1e-6)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) * 1.00390625)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -2.5e-5) || !(x <= 1.1e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -2.5e-5], N[Not[LessEqual[x, 1.1e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000012e-5 or 1.1000000000000001e-6 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 62.7%

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

      1. *-commutative 62.7%

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

   4. Simplified 62.7%

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

   if -2.50000000000000012e-5 < x < 1.1000000000000001e-6

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
   7. Step-by-step derivation

      1. fma-def 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      2. distribute-rgt1-in 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\sin y \cdot \left(0.00390625 + 1\right)\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot \color{blue}{1.00390625}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

   8. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00045 \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00045) (not (<= x 1.1e-6)))
     (/
      (+
       2.0
       (*
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))
        (+ (cos x) -1.0)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (fma -0.0625 (pow (sin y) 2.0) (* x (* (sin y) 1.00390625))))))
      (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00045) || !(x <= 1.1e-6)) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))) * (cos(x) + -1.0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) * 1.00390625)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00045) || !(x <= 1.1e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.00045], N[Not[LessEqual[x, 1.1e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999999e-4 or 1.1000000000000001e-6 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 62.7%

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

      1. *-commutative 62.7%

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

   4. Simplified 62.7%

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

   5. Taylor expanded in y around 0 59.9%

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

   if -4.4999999999999999e-4 < x < 1.1000000000000001e-6

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
   7. Step-by-step derivation

      1. fma-def 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      2. distribute-rgt1-in 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\sin y \cdot \left(0.00390625 + 1\right)\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot \color{blue}{1.00390625}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

   8. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00045 \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 12: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -4.5e-5) (not (<= x 1.1e-6)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
      (*
       3.0
       (+
        1.0
        (+
         (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
         (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0))))))))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (fma -0.0625 (pow (sin y) 2.0) (* x (* (sin y) 1.00390625))))))
      (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -4.5e-5) || !(x <= 1.1e-6)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * fma(-0.0625, pow(sin(y), 2.0), (x * (sin(y) * 1.00390625)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -4.5e-5) || !(x <= 1.1e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -4.5e-5], N[Not[LessEqual[x, 1.1e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000028e-5 or 1.1000000000000001e-6 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. metadata-eval 99.0%

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

      2. div-sub 99.0%

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

      3. div-inv 99.0%

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

      4. flip-- 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*l/ 98.9%

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

      7. sub-neg 98.9%

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

      8. metadata-eval 98.9%

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

      9. pow1/2 98.9%

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

      10. pow1/2 98.9%

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

      11. pow-sqr 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.3%

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

      1. associate-*r* 59.3%

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

   8. Simplified 59.3%

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

   if -4.50000000000000028e-5 < x < 1.1000000000000001e-6

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
   7. Step-by-step derivation

      1. fma-def 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      2. distribute-rgt1-in 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      3. *-commutative 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \color{blue}{\left(\sin y \cdot \left(0.00390625 + 1\right)\right)}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot \color{blue}{1.00390625}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]

   8. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)}\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right) - 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 13: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := 3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_0\right)\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2
         (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1)))))))
   (if (<= y -2.5e-6)
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) t_0)))) t_2)
     (if (<= y 2.3e-9)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (+
         3.0
         (+
          (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0)))
          (* 1.5 (- 3.0 (sqrt 5.0))))))
       (/ (+ 2.0 (* (- 1.0 (cos y)) (* (sqrt 2.0) (* -0.0625 t_0)))) t_2)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = 3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1))));
	double tmp;
	if (y <= -2.5e-6) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * t_0)))) / t_2;
	} else if (y <= 2.3e-9) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * 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 = sin(y) ** 2.0d0
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = 3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1))))
    if (y <= (-2.5d-6)) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * t_0)))) / t_2
    else if (y <= 2.3d-9) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * t_0)))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(y), 2.0);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = 3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1))));
	double tmp;
	if (y <= -2.5e-6) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * t_0)))) / t_2;
	} else if (y <= 2.3e-9) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * t_0)))) / t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(y), 2.0)
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = 3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1))))
	tmp = 0
	if y <= -2.5e-6:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * t_0)))) / t_2
	elif y <= 2.3e-9:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * t_0)))) / t_2
	return tmp

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1)))))
	tmp = 0.0
	if (y <= -2.5e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * t_0)))) / t_2);
	elseif (y <= 2.3e-9)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * t_0)))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) ^ 2.0;
	t_1 = sqrt(5.0) / 2.0;
	t_2 = 3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1))));
	tmp = 0.0;
	if (y <= -2.5e-6)
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * t_0)))) / t_2;
	elseif (y <= 2.3e-9)
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * t_0)))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[y, -2.5e-6], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 2.3e-9], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000002e-6

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 62.1%

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

   if -2.5000000000000002e-6 < y < 2.2999999999999999e-9

   1. Initial program 99.4%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if 2.2999999999999999e-9 < y

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 59.6%

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

   5. Taylor expanded in x around 0 59.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 14: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t_0 - 0.5\right)\\ t_2 := \sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot t_2}{3 \cdot \left(1 + \left(t_1 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot t_2}{3 \cdot \left(1 + \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (* (cos x) (- t_0 0.5)))
        (t_2 (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
   (if (<= y -2.4e-6)
     (/
      (+ 2.0 (* (- (cos x) (cos y)) t_2))
      (* 3.0 (+ 1.0 (+ t_1 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0))))))))
     (if (<= y 2.3e-9)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (+
         3.0
         (+
          (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0)))
          (* 1.5 (- 3.0 (sqrt 5.0))))))
       (/
        (+ 2.0 (* (- 1.0 (cos y)) t_2))
        (* 3.0 (+ 1.0 (+ t_1 (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(x) * (t_0 - 0.5);
	double t_2 = sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0));
	double tmp;
	if (y <= -2.4e-6) {
		tmp = (2.0 + ((cos(x) - cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	} else if (y <= 2.3e-9) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = cos(x) * (t_0 - 0.5d0)
    t_2 = sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))
    if (y <= (-2.4d-6)) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * t_2)) / (3.0d0 * (1.0d0 + (t_1 + (cos(y) * (2.0d0 / (3.0d0 + sqrt(5.0d0)))))))
    else if (y <= 2.3d-9) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * t_2)) / (3.0d0 * (1.0d0 + (t_1 + (cos(y) * (1.5d0 - t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.cos(x) * (t_0 - 0.5);
	double t_2 = Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0));
	double tmp;
	if (y <= -2.4e-6) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (Math.cos(y) * (2.0 / (3.0 + Math.sqrt(5.0)))))));
	} else if (y <= 2.3e-9) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (Math.cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(x) * (t_0 - 0.5)
	t_2 = math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))
	tmp = 0
	if y <= -2.4e-6:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (math.cos(y) * (2.0 / (3.0 + math.sqrt(5.0)))))))
	elif y <= 2.3e-9:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (math.cos(y) * (1.5 - t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(cos(x) * Float64(t_0 - 0.5))
	t_2 = Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0)))
	tmp = 0.0
	if (y <= -2.4e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * t_2)) / Float64(3.0 * Float64(1.0 + Float64(t_1 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0))))))));
	elseif (y <= 2.3e-9)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * t_2)) / Float64(3.0 * Float64(1.0 + Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = cos(x) * (t_0 - 0.5);
	t_2 = sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0));
	tmp = 0.0;
	if (y <= -2.4e-6)
		tmp = (2.0 + ((cos(x) - cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0)))))));
	elseif (y <= 2.3e-9)
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * t_2)) / (3.0 * (1.0 + (t_1 + (cos(y) * (1.5 - t_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[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e-6], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-9], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3999999999999999e-6

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

      1. metadata-eval 99.1%

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

      2. div-sub 99.1%

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

      3. div-inv 99.1%

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

      4. flip-- 99.0%

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

      5. metadata-eval 99.0%

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

      6. associate-*l/ 99.0%

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

      7. sub-neg 99.0%

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

      8. metadata-eval 99.0%

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

      9. pow1/2 99.0%

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

      10. pow1/2 99.0%

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

      11. pow-sqr 99.2%

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

      12. metadata-eval 99.2%

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

      13. metadata-eval 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

      16. metadata-eval 99.2%

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

      17. +-commutative 99.2%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 62.1%

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

   if -2.3999999999999999e-6 < y < 2.2999999999999999e-9

   1. Initial program 99.4%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if 2.2999999999999999e-9 < y

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 59.6%

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

   5. Taylor expanded in x around 0 59.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 15: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-6} \lor \neg \left(y \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= y -1.5e-6) (not (<= y 2.3e-9)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+
       3.0
       (+
        (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0)))
        (* 1.5 (- 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -1.5e-6) || !(y <= 2.3e-9)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((y <= (-1.5d-6)) .or. (.not. (y <= 2.3d-9))) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -1.5e-6) || !(y <= 2.3e-9)) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (y <= -1.5e-6) or not (y <= 2.3e-9):
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((y <= -1.5e-6) || !(y <= 2.3e-9))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((y <= -1.5e-6) || ~((y <= 2.3e-9)))
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.5e-6], N[Not[LessEqual[y, 2.3e-9]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e-6 or 2.2999999999999999e-9 < y

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 61.3%

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

   5. Taylor expanded in x around 0 60.7%

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

   if -1.5e-6 < y < 2.2999999999999999e-9

   1. Initial program 99.4%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 99.4%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 99.4%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-6} \lor \neg \left(y \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 16: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 + \left(6 \cdot \frac{\cos x}{t_1} + 1.5 \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot t_2, \frac{6}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_1 (+ (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -4.5e-6)
     (/ t_0 (+ 3.0 (+ (* 6.0 (/ (cos x) t_1)) (* 1.5 t_2))))
     (if (<= x 1.1e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (fma 1.5 (* (cos y) t_2) (/ 6.0 t_1))))
       (*
        0.3333333333333333
        (/
         t_0
         (+
          1.0
          (+
           (* 2.0 (/ 1.0 (+ 3.0 (sqrt 5.0))))
           (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_1 = sqrt(5.0) + 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4.5e-6) {
		tmp = t_0 / (3.0 + ((6.0 * (cos(x) / t_1)) + (1.5 * t_2)));
	} else if (x <= 1.1e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + fma(1.5, (cos(y) * t_2), (6.0 / t_1)));
	} else {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + ((2.0 * (1.0 / (3.0 + sqrt(5.0)))) + (cos(x) * ((sqrt(5.0) * 0.5) - 0.5)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_1 = Float64(sqrt(5.0) + 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4.5e-6)
		tmp = Float64(t_0 / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / t_1)) + Float64(1.5 * t_2))));
	elseif (x <= 1.1e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + fma(1.5, Float64(cos(y) * t_2), Float64(6.0 / t_1))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + Float64(Float64(2.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-6], N[(t$95$0 / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(6.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(N[(2.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000011e-6

   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 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 57.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if -4.50000000000000011e-6 < x < 1.1000000000000001e-6

   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 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. fma-def 99.2%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 6 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

      2. associate-*r/ 99.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), \color{blue}{\frac{6 \cdot 1}{1 + \sqrt{5}}}\right)} \]

      3. metadata-eval 99.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{\color{blue}{6}}{1 + \sqrt{5}}\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{6}{\color{blue}{\sqrt{5} + 1}}\right)} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{6}{\sqrt{5} + 1}\right)}} \]

   if 1.1000000000000001e-6 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.0%

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

      5. +-commutative 99.0%

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

   3. Simplified 98.9%

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

      1. metadata-eval 98.9%

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

      2. div-sub 98.9%

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

      3. div-inv 98.9%

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

      4. flip-- 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-*l/ 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

      9. pow1/2 98.8%

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

      10. pow1/2 98.8%

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

      11. pow-sqr 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in y around 0 58.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), \frac{6}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)}\\ \end{array} \]

Alternative 17: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + \cos x \cdot \left(t_0 - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))))
   (if (<= x -4.4e-6)
     (/
      t_1
      (+
       3.0
       (+ (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0))) (* 1.5 (- 3.0 (sqrt 5.0))))))
     (if (<= x 1.05e-6)
       (/
        (+
         2.0
         (* (- 1.0 (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (*
        0.3333333333333333
        (/
         t_1
         (+
          1.0
          (+ (* 2.0 (/ 1.0 (+ 3.0 (sqrt 5.0)))) (* (cos x) (- t_0 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double tmp;
	if (x <= -4.4e-6) {
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	} else if (x <= 1.05e-6) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((2.0 * (1.0 / (3.0 + sqrt(5.0)))) + (cos(x) * (t_0 - 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    if (x <= (-4.4d-6)) then
        tmp = t_1 / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    else if (x <= 1.05d-6) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - t_0))) - 0.5d0)))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + ((2.0d0 * (1.0d0 / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (t_0 - 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double tmp;
	if (x <= -4.4e-6) {
		tmp = t_1 / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	} else if (x <= 1.05e-6) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((2.0 * (1.0 / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (t_0 - 0.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	tmp = 0
	if x <= -4.4e-6:
		tmp = t_1 / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	elif x <= 1.05e-6:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((2.0 * (1.0 / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (t_0 - 0.5)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	tmp = 0.0
	if (x <= -4.4e-6)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	elseif (x <= 1.05e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(Float64(2.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(t_0 - 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	tmp = 0.0;
	if (x <= -4.4e-6)
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	elseif (x <= 1.05e-6)
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((2.0 * (1.0 / (3.0 + sqrt(5.0)))) + (cos(x) * (t_0 - 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-6], N[(t$95$1 / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-6], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(N[(2.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000002e-6

   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 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 57.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if -4.4000000000000002e-6 < x < 1.0499999999999999e-6

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 1.0499999999999999e-6 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.0%

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

      5. +-commutative 99.0%

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

   3. Simplified 98.9%

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

      1. metadata-eval 98.9%

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

      2. div-sub 98.9%

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

      3. div-inv 98.9%

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

      4. flip-- 98.8%

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

      5. metadata-eval 98.8%

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

      6. associate-*l/ 98.8%

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

      7. sub-neg 98.8%

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

      8. metadata-eval 98.8%

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

      9. pow1/2 98.8%

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

      10. pow1/2 98.8%

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

      11. pow-sqr 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

      14. metadata-eval 99.1%

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

      15. metadata-eval 99.1%

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

      16. metadata-eval 99.1%

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

      17. +-commutative 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in y around 0 58.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)}\\ \end{array} \]

Alternative 18: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} + 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{3 + \left(6 \cdot \frac{\cos x}{t_1} + 1.5 \cdot t_2\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot t_2\right) + 6 \cdot \frac{1}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_1 (+ (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -2.6e-5)
     (/ t_0 (+ 3.0 (+ (* 6.0 (/ (cos x) t_1)) (* 1.5 t_2))))
     (if (<= x 7.5e-7)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 1.5 (* (cos y) t_2)) (* 6.0 (/ 1.0 t_1)))))
       (*
        0.3333333333333333
        (/
         t_0
         (- (+ 2.5 (/ (cos x) (+ 0.5 (sqrt 1.25)))) (* (sqrt 5.0) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_1 = sqrt(5.0) + 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -2.6e-5) {
		tmp = t_0 / (3.0 + ((6.0 * (cos(x) / t_1)) + (1.5 * t_2)));
	} else if (x <= 7.5e-7) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((1.5 * (cos(y) * t_2)) + (6.0 * (1.0 / t_1))));
	} else {
		tmp = 0.3333333333333333 * (t_0 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - (sqrt(5.0) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    t_1 = sqrt(5.0d0) + 1.0d0
    t_2 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-2.6d-5)) then
        tmp = t_0 / (3.0d0 + ((6.0d0 * (cos(x) / t_1)) + (1.5d0 * t_2)))
    else if (x <= 7.5d-7) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((1.5d0 * (cos(y) * t_2)) + (6.0d0 * (1.0d0 / t_1))))
    else
        tmp = 0.3333333333333333d0 * (t_0 / ((2.5d0 + (cos(x) / (0.5d0 + sqrt(1.25d0)))) - (sqrt(5.0d0) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double t_1 = Math.sqrt(5.0) + 1.0;
	double t_2 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -2.6e-5) {
		tmp = t_0 / (3.0 + ((6.0 * (Math.cos(x) / t_1)) + (1.5 * t_2)));
	} else if (x <= 7.5e-7) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((1.5 * (Math.cos(y) * t_2)) + (6.0 * (1.0 / t_1))));
	} else {
		tmp = 0.3333333333333333 * (t_0 / ((2.5 + (Math.cos(x) / (0.5 + Math.sqrt(1.25)))) - (Math.sqrt(5.0) * 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	t_1 = math.sqrt(5.0) + 1.0
	t_2 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -2.6e-5:
		tmp = t_0 / (3.0 + ((6.0 * (math.cos(x) / t_1)) + (1.5 * t_2)))
	elif x <= 7.5e-7:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((1.5 * (math.cos(y) * t_2)) + (6.0 * (1.0 / t_1))))
	else:
		tmp = 0.3333333333333333 * (t_0 / ((2.5 + (math.cos(x) / (0.5 + math.sqrt(1.25)))) - (math.sqrt(5.0) * 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_1 = Float64(sqrt(5.0) + 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2.6e-5)
		tmp = Float64(t_0 / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / t_1)) + Float64(1.5 * t_2))));
	elseif (x <= 7.5e-7)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * t_2)) + Float64(6.0 * Float64(1.0 / t_1)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(Float64(2.5 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))) - Float64(sqrt(5.0) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	t_1 = sqrt(5.0) + 1.0;
	t_2 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -2.6e-5)
		tmp = t_0 / (3.0 + ((6.0 * (cos(x) / t_1)) + (1.5 * t_2)));
	elseif (x <= 7.5e-7)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((1.5 * (cos(y) * t_2)) + (6.0 * (1.0 / t_1))));
	else
		tmp = 0.3333333333333333 * (t_0 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - (sqrt(5.0) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-5], N[(t$95$0 / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-7], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$0 / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999984e-5

   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 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 57.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if -2.59999999999999984e-5 < x < 7.5000000000000002e-7

   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 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 99.5%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   if 7.5000000000000002e-7 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.0%

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

      5. +-commutative 99.0%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 58.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right) \cdot \cos x}\right) - 0.5 \cdot \sqrt{5}} \]

      2. flip-- 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}} \]

      3. associate-*l/ 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right) \cdot \cos x}{0.5 \cdot \sqrt{5} + 0.5}}\right) - 0.5 \cdot \sqrt{5}} \]

   6. Applied egg-rr 58.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\cos x}{0.5 + \sqrt{1.25}}}\right) - 0.5 \cdot \sqrt{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 6 \cdot \frac{1}{\sqrt{5} + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5}\\ \end{array} \]

Alternative 19: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))))
   (if (<= x -2.1e-5)
     (/
      t_1
      (+
       3.0
       (+ (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0))) (* 1.5 (- 3.0 (sqrt 5.0))))))
     (if (<= x 9e-7)
       (/
        (+
         2.0
         (* (- 1.0 (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (*
        0.3333333333333333
        (/ t_1 (- (+ 2.5 (/ (cos x) (+ 0.5 (sqrt 1.25)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double tmp;
	if (x <= -2.1e-5) {
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	} else if (x <= 9e-7) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    if (x <= (-2.1d-5)) then
        tmp = t_1 / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    else if (x <= 9d-7) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - t_0))) - 0.5d0)))
    else
        tmp = 0.3333333333333333d0 * (t_1 / ((2.5d0 + (cos(x) / (0.5d0 + sqrt(1.25d0)))) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double tmp;
	if (x <= -2.1e-5) {
		tmp = t_1 / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	} else if (x <= 9e-7) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (Math.cos(x) / (0.5 + Math.sqrt(1.25)))) - t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	tmp = 0
	if x <= -2.1e-5:
		tmp = t_1 / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	elif x <= 9e-7:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	else:
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (math.cos(x) / (0.5 + math.sqrt(1.25)))) - t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	tmp = 0.0
	if (x <= -2.1e-5)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	elseif (x <= 9e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(Float64(2.5 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	tmp = 0.0;
	if (x <= -2.1e-5)
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	elseif (x <= 9e-7)
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	else
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e-5], N[(t$95$1 / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-7], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999988e-5

   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 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 57.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if -2.09999999999999988e-5 < x < 8.99999999999999959e-7

   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 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 8.99999999999999959e-7 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.0%

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

      5. +-commutative 99.0%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 58.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right) \cdot \cos x}\right) - 0.5 \cdot \sqrt{5}} \]

      2. flip-- 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}} \]

      3. associate-*l/ 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right) \cdot \cos x}{0.5 \cdot \sqrt{5} + 0.5}}\right) - 0.5 \cdot \sqrt{5}} \]

   6. Applied egg-rr 58.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\cos x}{0.5 + \sqrt{1.25}}}\right) - 0.5 \cdot \sqrt{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5}\\ \end{array} \]

Alternative 20: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-6} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -5.2e-6) (not (<= x 1.1e-6)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (- (+ 2.5 (/ (cos x) (+ 0.5 (sqrt 1.25)))) t_0)))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 0.5 (+ t_0 (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -5.2e-6) || !(x <= 1.1e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-5.2d-6)) .or. (.not. (x <= 1.1d-6))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((2.5d0 + (cos(x) / (0.5d0 + sqrt(1.25d0)))) - t_0))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (t_0 + (2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -5.2e-6) || !(x <= 1.1e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((2.5 + (Math.cos(x) / (0.5 + Math.sqrt(1.25)))) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (0.5 + (t_0 + (2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -5.2e-6) or not (x <= 1.1e-6):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((2.5 + (math.cos(x) / (0.5 + math.sqrt(1.25)))) - t_0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (0.5 + (t_0 + (2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -5.2e-6) || !(x <= 1.1e-6))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(2.5 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))) - t_0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(t_0 + Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -5.2e-6) || ~((x <= 1.1e-6)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -5.2e-6], N[Not[LessEqual[x, 1.1e-6]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(t$95$0 + N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.20000000000000019e-6 or 1.1000000000000001e-6 < x

   1. Initial program 99.0%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.1%

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

      5. +-commutative 99.1%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 58.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 58.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right) \cdot \cos x}\right) - 0.5 \cdot \sqrt{5}} \]

      2. flip-- 58.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}} \]

      3. associate-*l/ 58.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right) \cdot \cos x}{0.5 \cdot \sqrt{5} + 0.5}}\right) - 0.5 \cdot \sqrt{5}} \]

   6. Applied egg-rr 58.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\cos x}{0.5 + \sqrt{1.25}}}\right) - 0.5 \cdot \sqrt{5}} \]

   if -5.20000000000000019e-6 < x < 1.1000000000000001e-6

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*l* 99.6%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

   3. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. div-sub 99.7%

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

      3. div-inv 99.7%

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

      4. flip-- 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. pow1/2 99.6%

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

      10. pow1/2 99.6%

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

      11. pow-sqr 99.7%

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

      12. metadata-eval 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-6} \lor \neg \left(x \leq 1.1 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 21: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))))
   (if (<= x -7.4e-6)
     (/
      t_1
      (+
       3.0
       (+ (* 6.0 (/ (cos x) (+ (sqrt 5.0) 1.0))) (* 1.5 (- 3.0 (sqrt 5.0))))))
     (if (<= x 1.1e-6)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 0.5 (+ t_0 (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))
       (*
        0.3333333333333333
        (/ t_1 (- (+ 2.5 (/ (cos x) (+ 0.5 (sqrt 1.25)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double tmp;
	if (x <= -7.4e-6) {
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	} else if (x <= 1.1e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    if (x <= (-7.4d-6)) then
        tmp = t_1 / (3.0d0 + ((6.0d0 * (cos(x) / (sqrt(5.0d0) + 1.0d0))) + (1.5d0 * (3.0d0 - sqrt(5.0d0)))))
    else if (x <= 1.1d-6) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (t_0 + (2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))))
    else
        tmp = 0.3333333333333333d0 * (t_1 / ((2.5d0 + (cos(x) / (0.5d0 + sqrt(1.25d0)))) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double tmp;
	if (x <= -7.4e-6) {
		tmp = t_1 / (3.0 + ((6.0 * (Math.cos(x) / (Math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - Math.sqrt(5.0)))));
	} else if (x <= 1.1e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (0.5 + (t_0 + (2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (Math.cos(x) / (0.5 + Math.sqrt(1.25)))) - t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	tmp = 0
	if x <= -7.4e-6:
		tmp = t_1 / (3.0 + ((6.0 * (math.cos(x) / (math.sqrt(5.0) + 1.0))) + (1.5 * (3.0 - math.sqrt(5.0)))))
	elif x <= 1.1e-6:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (0.5 + (t_0 + (2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))))
	else:
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (math.cos(x) / (0.5 + math.sqrt(1.25)))) - t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	tmp = 0.0
	if (x <= -7.4e-6)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(x) / Float64(sqrt(5.0) + 1.0))) + Float64(1.5 * Float64(3.0 - sqrt(5.0))))));
	elseif (x <= 1.1e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(t_0 + Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(Float64(2.5 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	tmp = 0.0;
	if (x <= -7.4e-6)
		tmp = t_1 / (3.0 + ((6.0 * (cos(x) / (sqrt(5.0) + 1.0))) + (1.5 * (3.0 - sqrt(5.0)))));
	elseif (x <= 1.1e-6)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	else
		tmp = 0.3333333333333333 * (t_1 / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.4e-6], N[(t$95$1 / N[(3.0 + N[(N[(6.0 * N[(N[Cos[x], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(t$95$0 + N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4000000000000003e-6

   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 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. div-inv 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + -1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} + \color{blue}{\left(-1\right)}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      4. sub-neg 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} - 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      5. flip-- 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} - \color{blue}{-1 \cdot -1}}{\sqrt{5} + 1}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      8. associate-/r/ 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - -1 \cdot -1} \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      9. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\sqrt{5} \cdot \sqrt{5} - \color{blue}{1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      10. sub-neg 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{\sqrt{5} \cdot \sqrt{5} + \left(-1\right)}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      11. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{0.5}} \cdot \sqrt{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      12. pow1/2 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{0.5} \cdot \color{blue}{{5}^{0.5}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      13. pow-sqr 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{{5}^{\left(2 \cdot 0.5\right)}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      14. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{{5}^{\color{blue}{1}} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      15. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{5} + \left(-1\right)} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      16. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{5 + \color{blue}{-1}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      17. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\frac{0.6666666666666666}{\color{blue}{4}} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      18. metadata-eval 98.9%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{1}{\color{blue}{0.16666666666666666} \cdot \left(\sqrt{5} + 1\right)} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{1}{0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. associate-/r* 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \color{blue}{\frac{\frac{1}{0.16666666666666666}}{\sqrt{5} + 1}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{\color{blue}{6}}{\sqrt{5} + 1} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

      3. +-commutative 98.9%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\cos x \cdot \frac{6}{\color{blue}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{\cos x \cdot \frac{6}{1 + \sqrt{5}}} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 57.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(3 - \sqrt{5}\right) + 6 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)}} \]

   if -7.4000000000000003e-6 < x < 1.1000000000000001e-6

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*l* 99.6%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

   3. Simplified 99.7%

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

      1. metadata-eval 99.7%

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

      2. div-sub 99.7%

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

      3. div-inv 99.7%

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

      4. flip-- 99.6%

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

      5. metadata-eval 99.6%

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

      6. associate-*l/ 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. pow1/2 99.6%

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

      10. pow1/2 99.6%

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

      11. pow-sqr 99.7%

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

      12. metadata-eval 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. metadata-eval 99.7%

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

      17. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.1000000000000001e-6 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-def 99.0%

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

      5. +-commutative 99.0%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 58.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 58.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right) \cdot \cos x}\right) - 0.5 \cdot \sqrt{5}} \]

      2. flip-- 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}} \]

      3. associate-*l/ 58.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right) \cdot \cos x}{0.5 \cdot \sqrt{5} + 0.5}}\right) - 0.5 \cdot \sqrt{5}} \]

   6. Applied egg-rr 58.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\cos x}{0.5 + \sqrt{1.25}}}\right) - 0.5 \cdot \sqrt{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos x}{\sqrt{5} + 1} + 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5}\\ \end{array} \]

Alternative 22: 60.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
   (- (+ 2.5 (/ (cos x) (+ 0.5 (sqrt 1.25)))) (* (sqrt 5.0) 0.5)))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - (sqrt(5.0) * 0.5)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((2.5d0 + (cos(x) / (0.5d0 + sqrt(1.25d0)))) - (sqrt(5.0d0) * 0.5d0)))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((2.5 + (Math.cos(x) / (0.5 + Math.sqrt(1.25)))) - (Math.sqrt(5.0) * 0.5)));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((2.5 + (math.cos(x) / (0.5 + math.sqrt(1.25)))) - (math.sqrt(5.0) * 0.5)))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(2.5 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))) - Float64(sqrt(5.0) * 0.5))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + (cos(x) / (0.5 + sqrt(1.25)))) - (sqrt(5.0) * 0.5)));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $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-/r* 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
5. Step-by-step derivation

   1. *-commutative 59.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\left(0.5 \cdot \sqrt{5} - 0.5\right) \cdot \cos x}\right) - 0.5 \cdot \sqrt{5}} \]

   2. flip-- 59.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5}{0.5 \cdot \sqrt{5} + 0.5}} \cdot \cos x\right) - 0.5 \cdot \sqrt{5}} \]

   3. associate-*l/ 59.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\left(\left(0.5 \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot \sqrt{5}\right) - 0.5 \cdot 0.5\right) \cdot \cos x}{0.5 \cdot \sqrt{5} + 0.5}}\right) - 0.5 \cdot \sqrt{5}} \]

6. Applied egg-rr 59.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \color{blue}{\frac{\cos x}{0.5 + \sqrt{1.25}}}\right) - 0.5 \cdot \sqrt{5}} \]

7. Final simplification 59.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right) - \sqrt{5} \cdot 0.5} \]

Alternative 23: 60.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
   (+ (* (cos x) (+ -0.5 (sqrt 1.25))) (- 2.5 (sqrt 1.25))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((cos(x) * ((-0.5d0) + sqrt(1.25d0))) + (2.5d0 - sqrt(1.25d0))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((Math.cos(x) * (-0.5 + Math.sqrt(1.25))) + (2.5 - Math.sqrt(1.25))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((math.cos(x) * (-0.5 + math.sqrt(1.25))) + (2.5 - math.sqrt(1.25))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(cos(x) * Float64(-0.5 + sqrt(1.25))) + Float64(2.5 - sqrt(1.25)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.5 - N[Sqrt[1.25], $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-/r* 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
5. Step-by-step derivation

   1. +-commutative 59.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} - 0.5 \cdot \sqrt{5}} \]

   2. associate--l+ 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \left(2.5 - 0.5 \cdot \sqrt{5}\right)}} \]

6. Applied egg-rr 59.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)}} \]

7. Final simplification 59.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)} \]

Alternative 24: 40.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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-/r* 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]

5. Taylor expanded in x around 0 40.5%

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

6. Final simplification 40.5%

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

Alternative 25: 40.8% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

   1. associate-/r* 99.3%

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

   2. +-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-def 99.3%

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

   5. +-commutative 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]

5. Taylor expanded in x around 0 40.5%

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

6. Taylor expanded in x around 0 32.1%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(-0.5 \cdot \left({x}^{4} \cdot \sqrt{2}\right)\right)}}{2} \]

7. Taylor expanded in x around 0 40.5%

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

8. Final simplification 40.5%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023326 
(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))))))