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

Percentage Accurate: 99.3% → 99.3%

Time: 50.7s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(3.0 + N[(N[(N[Cos[x], $MachinePrecision] * 1.5), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   4. pow1/2 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

   2. associate-*l/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. +-commutative 99.4%

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

    2. associate-*r* 99.4%

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

    3. fma-define 99.4%

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

    4. associate-*r/ 99.4%

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

12. Applied egg-rr 99.4%

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

    1. div-inv 99.4%

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

    2. +-commutative 99.4%

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

    3. fma-define 99.4%

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

    4. +-commutative 99.4%

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

    5. fma-define 99.4%

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

    6. *-commutative 99.4%

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

    7. *-un-lft-identity 99.4%

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

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

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

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_2}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_2\right) + 1.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.0039)
     (/
      (+
       2.0
       (*
        (+ (cos x) -1.0)
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.0031)
       (/
        (fma
         (sqrt 2.0)
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (+ (sin x) (* -0.0625 (sin y))) (- 1.0 (cos y))))
         2.0)
        (+ 3.0 (+ (* 1.5 (* (cos x) t_2)) (* 1.5 (* (cos y) t_0)))))
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.0039) {
		tmp = (2.0 + ((cos(x) + -1.0) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.0031) {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (1.0 - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(x) * t_2)) + (1.5 * (cos(y) * t_0))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.0039)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.0031)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_2)) + Float64(1.5 * Float64(cos(y) * t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.0039], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0031], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0038999999999999998

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 61.4%

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

   if -0.0038999999999999998 < x < 0.00309999999999999989

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 0.00309999999999999989 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0039:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0031:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_1\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.0102)
     (/
      (+
       2.0
       (*
        (+ (cos x) -1.0)
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_1 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 0.0055)
       (/
        (fma
         (sqrt 2.0)
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (+ (sin x) (* -0.0625 (sin y))) (- 1.0 (cos y))))
         2.0)
        (+
         3.0
         (+ (* 1.5 (* (cos x) t_1)) (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.0102) {
		tmp = (2.0 + ((cos(x) + -1.0) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 0.0055) {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (1.0 - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(x) * t_1)) + (cos(y) * (6.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.0102)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 0.0055)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_1)) + Float64(cos(y) * Float64(6.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 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))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.0102], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 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], If[LessEqual[x, 0.0055], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.010200000000000001

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 61.4%

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

   if -0.010200000000000001 < x < 0.0054999999999999997

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 99.7%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0102:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x + -0.0625 \cdot \sin y\\ t_1 := 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ t_2 := \sin y + \sin x \cdot -0.0625\\ t_3 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_2 \cdot \left(t\_0 \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(t\_1 + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_2 \cdot \left(t\_0 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(t\_1 + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_3 - 0.5\right) + \cos y \cdot \left(1.5 - t\_3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sin x) (* -0.0625 (sin y))))
        (t_1 (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0))))
        (t_2 (+ (sin y) (* (sin x) -0.0625)))
        (t_3 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.0055)
     (/
      (fma (sqrt 2.0) (* t_2 (* t_0 (+ (cos x) -1.0))) 2.0)
      (+ 3.0 (+ t_1 (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0)))))))
     (if (<= x 0.0072)
       (/
        (fma (sqrt 2.0) (* t_2 (* t_0 (- 1.0 (cos y)))) 2.0)
        (+ 3.0 (+ t_1 (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (* (cos y) (- 1.5 t_3))))))))))

C

double code(double x, double y) {
	double t_0 = sin(x) + (-0.0625 * sin(y));
	double t_1 = 1.5 * (cos(x) * (sqrt(5.0) + -1.0));
	double t_2 = sin(y) + (sin(x) * -0.0625);
	double t_3 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.0055) {
		tmp = fma(sqrt(2.0), (t_2 * (t_0 * (cos(x) + -1.0))), 2.0) / (3.0 + (t_1 + (1.5 * (cos(y) * (3.0 - sqrt(5.0))))));
	} else if (x <= 0.0072) {
		tmp = fma(sqrt(2.0), (t_2 * (t_0 * (1.0 - cos(y)))), 2.0) / (3.0 + (t_1 + (cos(y) * (6.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) + Float64(-0.0625 * sin(y)))
	t_1 = Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))
	t_2 = Float64(sin(y) + Float64(sin(x) * -0.0625))
	t_3 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.0055)
		tmp = Float64(fma(sqrt(2.0), Float64(t_2 * Float64(t_0 * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(t_1 + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 0.0072)
		tmp = Float64(fma(sqrt(2.0), Float64(t_2 * Float64(t_0 * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(t_1 + Float64(cos(y) * Float64(6.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_3))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.0055], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(t$95$1 + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0072], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(t$95$0 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0054999999999999997

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 99.1%

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

   6. Taylor expanded in y around 0 61.4%

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

   if -0.0054999999999999997 < x < 0.0071999999999999998

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 99.7%

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

   if 0.0071999999999999998 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0055:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x + -0.0625 \cdot \sin y\\ t_1 := 3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \sin y + \sin x \cdot -0.0625\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_3 \cdot \left(t\_0 \cdot \left(\cos x + -1\right)\right), 2\right)}{t\_1}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_3 \cdot \left(t\_0 \cdot \left(1 - \cos y\right)\right), 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_2 - 0.5\right) + \cos y \cdot \left(1.5 - t\_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sin x) (* -0.0625 (sin y))))
        (t_1
         (+
          3.0
          (+
           (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))
           (* (cos y) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (+ (sin y) (* (sin x) -0.0625))))
   (if (<= x -0.0023)
     (/ (fma (sqrt 2.0) (* t_3 (* t_0 (+ (cos x) -1.0))) 2.0) t_1)
     (if (<= x 0.0032)
       (/ (fma (sqrt 2.0) (* t_3 (* t_0 (- 1.0 (cos y)))) 2.0) t_1)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))))))

C

double code(double x, double y) {
	double t_0 = sin(x) + (-0.0625 * sin(y));
	double t_1 = 3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (cos(y) * (6.0 / (3.0 + sqrt(5.0)))));
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = sin(y) + (sin(x) * -0.0625);
	double tmp;
	if (x <= -0.0023) {
		tmp = fma(sqrt(2.0), (t_3 * (t_0 * (cos(x) + -1.0))), 2.0) / t_1;
	} else if (x <= 0.0032) {
		tmp = fma(sqrt(2.0), (t_3 * (t_0 * (1.0 - cos(y)))), 2.0) / t_1;
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) + Float64(-0.0625 * sin(y)))
	t_1 = Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(cos(y) * Float64(6.0 / Float64(3.0 + sqrt(5.0))))))
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = Float64(sin(y) + Float64(sin(x) * -0.0625))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(fma(sqrt(2.0), Float64(t_3 * Float64(t_0 * Float64(cos(x) + -1.0))), 2.0) / t_1);
	elseif (x <= 0.0032)
		tmp = Float64(fma(sqrt(2.0), Float64(t_3 * Float64(t_0 * Float64(1.0 - cos(y)))), 2.0) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 0.0032], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * N[(t$95$0 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0023

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   8. Simplified 99.2%

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

      1. fma-undefine 99.2%

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

      2. associate-*l/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

   10. Applied egg-rr 99.2%

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

   11. Taylor expanded in y around 0 61.6%

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

   if -0.0023 < x < 0.00320000000000000015

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 99.7%

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

   if 0.00320000000000000015 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \cos y \cdot \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

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) + (-0.0625 * sin(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
}

Fortran

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

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) + (-0.0625 * math.sin(y)))))) / (3.0 + ((1.5 * (math.cos(x) * (math.sqrt(5.0) + -1.0))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))))

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(-0.0625 * sin(y)))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))))
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) + (-0.0625 * sin(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
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[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   4. pow1/2 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

   2. associate-*l/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. +-commutative 99.4%

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

    2. associate-*r* 99.4%

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

    3. fma-define 99.4%

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

    4. associate-*r/ 99.4%

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

12. Applied egg-rr 99.4%

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

13. Taylor expanded in y around inf 99.4%

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

14. Final simplification 99.4%

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

Alternative 7: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.0003 \lor \neg \left(y \leq 0.00072\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= y -0.0003) (not (<= y 0.00072)))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (+ (sin x) (* -0.0625 (sin y))) (+ (cos x) -1.0)))
       2.0)
      (+
       3.0
       (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (- 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -0.0003) || !(y <= 0.00072)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) + -1.0))), 2.0) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.0003) || !(y <= 0.00072))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - 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(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(3.0 - sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0003], N[Not[LessEqual[y, 0.00072]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - 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[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.99999999999999974e-4 or 7.20000000000000045e-4 < y

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-rgt-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-rgt-in 99.1%

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

      5. associate-+l+ 99.2%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 60.2%

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

   if -2.99999999999999974e-4 < y < 7.20000000000000045e-4

   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%

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

   5. Taylor expanded in y around inf 99.5%

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

   6. Taylor expanded in y around 0 99.2%

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

   7. Taylor expanded in y around 0 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0003 \lor \neg \left(y \leq 0.00072\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x - \frac{\sin y}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 1 - \cos y\\ t_5 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;\frac{2 + t\_4 \cdot \left(\left(\sqrt{2} \cdot t\_0\right) \cdot t\_3\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_5}{2}\right) + \cos y \cdot \frac{t\_2}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00015:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot t\_5 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_4 \cdot \left(\sqrt{2} \cdot \left(t\_0 \cdot t\_3\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin x) (/ (sin y) 16.0)))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (- 1.0 (cos y)))
        (t_5 (+ (sqrt 5.0) -1.0)))
   (if (<= y -0.00075)
     (/
      (+ 2.0 (* t_4 (* (* (sqrt 2.0) t_0) t_3)))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_5 2.0))) (* (cos y) (/ t_2 2.0)))))
     (if (<= y 0.00015)
       (/
        (fma
         (sqrt 2.0)
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (+ (sin x) (* -0.0625 (sin y))) (+ (cos x) -1.0)))
         2.0)
        (+ 3.0 (* 1.5 (+ (* (cos x) t_5) t_2))))
       (/
        (+ 2.0 (* t_4 (* (sqrt 2.0) (* t_0 t_3))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = sin(x) - (sin(y) / 16.0);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = 1.0 - cos(y);
	double t_5 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -0.00075) {
		tmp = (2.0 + (t_4 * ((sqrt(2.0) * t_0) * t_3))) / (3.0 * ((1.0 + (cos(x) * (t_5 / 2.0))) + (cos(y) * (t_2 / 2.0))));
	} else if (y <= 0.00015) {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) + -1.0))), 2.0) / (3.0 + (1.5 * ((cos(x) * t_5) + t_2)));
	} else {
		tmp = (2.0 + (t_4 * (sqrt(2.0) * (t_0 * t_3)))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) - Float64(sin(y) / 16.0))
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(1.0 - cos(y))
	t_5 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -0.00075)
		tmp = Float64(Float64(2.0 + Float64(t_4 * Float64(Float64(sqrt(2.0) * t_0) * t_3))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_5 / 2.0))) + Float64(cos(y) * Float64(t_2 / 2.0)))));
	elseif (y <= 0.00015)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_5) + t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_4 * Float64(sqrt(2.0) * Float64(t_0 * t_3)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -0.00075], N[(N[(2.0 + N[(t$95$4 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$5 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00015], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$5), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5000000000000002e-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. Add Preprocessing

   3. Taylor expanded in x around 0 56.7%

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

   if -7.5000000000000002e-4 < y < 1.49999999999999987e-4

   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%

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

   5. Taylor expanded in y around inf 99.5%

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

   6. Taylor expanded in y around 0 99.2%

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

   7. Taylor expanded in y around 0 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

   if 1.49999999999999987e-4 < y

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 99.1%

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

      5. associate-+l+ 99.2%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00075:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00015:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x - \frac{\sin y}{16}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot t\_0\right) \cdot t\_3\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00023:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(t\_0 \cdot t\_3\right)\right)}{3 \cdot \left(1 + \left(\left(t\_1 + \cos y \cdot \left(1.5 - t\_1\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_2 - 0.5\right) + \cos y \cdot \left(1.5 - t\_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin x) (/ (sin y) 16.0)))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -5.4e-5)
     (/
      (+ 2.0 (* (+ (cos x) -1.0) (* (* (sqrt 2.0) t_0) t_3)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 0.00023)
       (/
        (+ 2.0 (* (- 1.0 (cos y)) (* (sqrt 2.0) (* t_0 t_3))))
        (* 3.0 (+ 1.0 (- (+ t_1 (* (cos y) (- 1.5 t_1))) 0.5))))
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))))))

C

double code(double x, double y) {
	double t_0 = sin(x) - (sin(y) / 16.0);
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -5.4e-5) {
		tmp = (2.0 + ((cos(x) + -1.0) * ((sqrt(2.0) * t_0) * t_3))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 0.00023) {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (t_0 * t_3)))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - 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) :: t_3
    real(8) :: tmp
    t_0 = sin(x) - (sin(y) / 16.0d0)
    t_1 = sqrt(5.0d0) * 0.5d0
    t_2 = sqrt(5.0d0) / 2.0d0
    t_3 = sin(y) - (sin(x) / 16.0d0)
    if (x <= (-5.4d-5)) then
        tmp = (2.0d0 + ((cos(x) + (-1.0d0)) * ((sqrt(2.0d0) * t_0) * t_3))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (x <= 0.00023d0) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (t_0 * t_3)))) / (3.0d0 * (1.0d0 + ((t_1 + (cos(y) * (1.5d0 - t_1))) - 0.5d0)))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + (cos(y) * (1.5d0 - t_2)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) - (Math.sin(y) / 16.0);
	double t_1 = Math.sqrt(5.0) * 0.5;
	double t_2 = Math.sqrt(5.0) / 2.0;
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if (x <= -5.4e-5) {
		tmp = (2.0 + ((Math.cos(x) + -1.0) * ((Math.sqrt(2.0) * t_0) * t_3))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (x <= 0.00023) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * (t_0 * t_3)))) / (3.0 * (1.0 + ((t_1 + (Math.cos(y) * (1.5 - t_1))) - 0.5)));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) - (math.sin(y) / 16.0)
	t_1 = math.sqrt(5.0) * 0.5
	t_2 = math.sqrt(5.0) / 2.0
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if x <= -5.4e-5:
		tmp = (2.0 + ((math.cos(x) + -1.0) * ((math.sqrt(2.0) * t_0) * t_3))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif x <= 0.00023:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * (t_0 * t_3)))) / (3.0 * (1.0 + ((t_1 + (math.cos(y) * (1.5 - t_1))) - 0.5)))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_2 - 0.5)) + (math.cos(y) * (1.5 - t_2)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) - Float64(sin(y) / 16.0))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -5.4e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(Float64(sqrt(2.0) * t_0) * t_3))) / 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)))));
	elseif (x <= 0.00023)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(t_0 * t_3)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_1))) - 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) - (sin(y) / 16.0);
	t_1 = sqrt(5.0) * 0.5;
	t_2 = sqrt(5.0) / 2.0;
	t_3 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if (x <= -5.4e-5)
		tmp = (2.0 + ((cos(x) + -1.0) * ((sqrt(2.0) * t_0) * t_3))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (x <= 0.00023)
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * (t_0 * t_3)))) / (3.0 * (1.0 + ((t_1 + (cos(y) * (1.5 - t_1))) - 0.5)));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.4e-5], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$3), $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], If[LessEqual[x, 0.00023], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3999999999999998e-5

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 61.4%

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

   if -5.3999999999999998e-5 < x < 2.3000000000000001e-4

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. cos-neg 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 2.3000000000000001e-4 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00023:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_3 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.00043:\\ \;\;\;\;\frac{2 + t\_1 \cdot t\_2}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(t\_0 - 0.5\right)\right) - t\_0\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot t\_2}{3 \cdot \left(1 + \left(\left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_3 - 0.5\right) + \cos y \cdot \left(1.5 - t\_3\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (*
          (sqrt 2.0)
          (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0)))))
        (t_3 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.00043)
     (/
      (+ 2.0 (* t_1 t_2))
      (* 3.0 (+ 1.0 (- (+ 1.5 (* (cos x) (- t_0 0.5))) t_0))))
     (if (<= x 2.55e-5)
       (/
        (+ 2.0 (* (- 1.0 (cos y)) t_2))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (/
        (+ 2.0 (* t_1 (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (* (cos y) (- 1.5 t_3))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)));
	double t_3 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.00043) {
		tmp = (2.0 + (t_1 * t_2)) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	} else if (x <= 2.55e-5) {
		tmp = (2.0 + ((1.0 - cos(y)) * t_2)) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = (2.0 + (t_1 * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = cos(x) - cos(y)
    t_2 = sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))
    t_3 = sqrt(5.0d0) / 2.0d0
    if (x <= (-0.00043d0)) then
        tmp = (2.0d0 + (t_1 * t_2)) / (3.0d0 * (1.0d0 + ((1.5d0 + (cos(x) * (t_0 - 0.5d0))) - t_0)))
    else if (x <= 2.55d-5) then
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * t_2)) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - t_0))) - 0.5d0)))
    else
        tmp = (2.0d0 + (t_1 * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_3 - 0.5d0)) + (cos(y) * (1.5d0 - t_3)))))
    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 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * (Math.sin(y) - (Math.sin(x) / 16.0)));
	double t_3 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.00043) {
		tmp = (2.0 + (t_1 * t_2)) / (3.0 * (1.0 + ((1.5 + (Math.cos(x) * (t_0 - 0.5))) - t_0)));
	} else if (x <= 2.55e-5) {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * t_2)) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = (2.0 + (t_1 * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_3 - 0.5)) + (Math.cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * (math.sin(y) - (math.sin(x) / 16.0)))
	t_3 = math.sqrt(5.0) / 2.0
	tmp = 0
	if x <= -0.00043:
		tmp = (2.0 + (t_1 * t_2)) / (3.0 * (1.0 + ((1.5 + (math.cos(x) * (t_0 - 0.5))) - t_0)))
	elif x <= 2.55e-5:
		tmp = (2.0 + ((1.0 - math.cos(y)) * t_2)) / (3.0 * (1.0 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	else:
		tmp = (2.0 + (t_1 * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_3 - 0.5)) + (math.cos(y) * (1.5 - t_3)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))))
	t_3 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.00043)
		tmp = Float64(Float64(2.0 + Float64(t_1 * t_2)) / Float64(3.0 * Float64(1.0 + Float64(Float64(1.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0))));
	elseif (x <= 2.55e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * t_2)) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_3))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = cos(x) - cos(y);
	t_2 = sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)));
	t_3 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (x <= -0.00043)
		tmp = (2.0 + (t_1 * t_2)) / (3.0 * (1.0 + ((1.5 + (cos(x) * (t_0 - 0.5))) - t_0)));
	elseif (x <= 2.55e-5)
		tmp = (2.0 + ((1.0 - cos(y)) * t_2)) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	else
		tmp = (2.0 + (t_1 * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_3 - 0.5)) + (cos(y) * (1.5 - t_3)))));
	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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.00043], N[(N[(2.0 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-5], 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[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.29999999999999989e-4

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 60.2%

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

   if -4.29999999999999989e-4 < x < 2.54999999999999998e-5

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. cos-neg 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 2.54999999999999998e-5 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in y around 0 63.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00043:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\left(1.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.000235 \lor \neg \left(y \leq 0.000122\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= y -0.000235) (not (<= y 0.000122)))
     (/
      (+
       2.0
       (* (- (cos x) (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))))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (+ (sin x) (* -0.0625 (sin y))) (+ (cos x) -1.0)))
       2.0)
      (+
       3.0
       (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (- 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((y <= -0.000235) || !(y <= 0.000122)) {
		tmp = (2.0 + ((cos(x) - 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 = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) + -1.0))), 2.0) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.000235) || !(y <= 0.000122))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - 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(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(3.0 - sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.000235], N[Not[LessEqual[y, 0.000122]], $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[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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.34999999999999993e-4 or 1.21999999999999997e-4 < y

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-rgt-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-rgt-in 99.1%

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

      5. associate-+l+ 99.2%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 60.0%

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

      1. associate-*r* 60.0%

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

   7. Simplified 60.0%

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

   if -2.34999999999999993e-4 < y < 1.21999999999999997e-4

   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%

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

   5. Taylor expanded in y around inf 99.5%

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

   6. Taylor expanded in y around 0 99.2%

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

   7. Taylor expanded in y around 0 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000235 \lor \neg \left(y \leq 0.000122\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -4.4 \lor \neg \left(x \leq 0.0046\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -4.4) (not (<= x 0.0046)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (+ (* -0.0625 (pow (sin y) 2.0)) (* x (* (sin y) 1.00390625))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -4.4) || !(x <= 0.0046)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * pow(sin(y), 2.0)) + (x * (sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-4.4d0)) .or. (.not. (x <= 0.0046d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (((-0.0625d0) * (sin(y) ** 2.0d0)) + (x * (sin(y) * 1.00390625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -4.4) || !(x <= 0.0046)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * ((-0.0625 * Math.pow(Math.sin(y), 2.0)) + (x * (Math.sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -4.4) or not (x <= 0.0046):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (math.sqrt(2.0) * ((-0.0625 * math.pow(math.sin(y), 2.0)) + (x * (math.sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -4.4) || !(x <= 0.0046))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 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(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) + Float64(x * Float64(sin(y) * 1.00390625)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -4.4) || ~((x <= 0.0046)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((-0.0625 * (sin(y) ^ 2.0)) + (x * (sin(y) * 1.00390625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -4.4], N[Not[LessEqual[x, 0.0046]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000004 or 0.0045999999999999999 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.5%

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

   if -4.4000000000000004 < x < 0.0045999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.0%

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

   4. Taylor expanded in x around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*l* 99.0%

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

      6. distribute-lft-out 99.0%

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

      7. distribute-rgt1-in 99.0%

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

      8. *-commutative 99.0%

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

      9. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \lor \neg \left(x \leq 0.0046\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2} + x \cdot \left(\sin y \cdot 1.00390625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5} \lor \neg \left(x \leq 0.00019\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -2.5e-5) (not (<= x 0.00019)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))
     (/
      (+
       2.0
       (*
        (- 1.0 (cos y))
        (*
         (sqrt 2.0)
         (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))))
      (* 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 t_1 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -2.5e-5) || !(x <= 0.00019)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - 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 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-2.5d-5)) .or. (.not. (x <= 0.00019d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - 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 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -2.5e-5) || !(x <= 0.00019)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	} else {
		tmp = (2.0 + ((1.0 - 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 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -2.5e-5) or not (x <= 0.00019):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	else:
		tmp = (2.0 + ((1.0 - 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 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -2.5e-5) || !(x <= 0.00019))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - 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(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -2.5e-5) || ~((x <= 0.00019)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - 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[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -2.5e-5], N[Not[LessEqual[x, 0.00019]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(1.0 - 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[(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 1.9000000000000001e-4 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.2%

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

   if -2.50000000000000012e-5 < x < 1.9000000000000001e-4

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. cos-neg 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in x around 0 99.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5} \lor \neg \left(x \leq 0.00019\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6} \lor \neg \left(x \leq 1.16 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -6.2e-6) (not (<= x 1.16e-5)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+
       3.0
       (+
        (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
        (* 1.5 (+ (sqrt 5.0) -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -6.2e-6) || !(x <= 1.16e-5)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 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(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-6.2d-6)) .or. (.not. (x <= 1.16d-5))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 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(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (1.5d0 * (sqrt(5.0d0) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -6.2e-6) || !(x <= 1.16e-5)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 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(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (1.5 * (Math.sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -6.2e-6) or not (x <= 1.16e-5):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 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(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (1.5 * (math.sqrt(5.0) + -1.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -6.2e-6) || !(x <= 1.16e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 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(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(sqrt(5.0) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -6.2e-6) || ~((x <= 1.16e-5)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 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(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -6.2e-6], N[Not[LessEqual[x, 1.16e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * 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[(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[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[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999999e-6 or 1.1600000000000001e-5 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.2%

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

   if -6.1999999999999999e-6 < x < 1.1600000000000001e-5

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. associate-*r* 99.7%

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

       3. fma-define 99.7%

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

       4. associate-*r/ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in x around 0 99.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-6} \lor \neg \left(x \leq 1.16 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.000235 \lor \neg \left(y \leq 1.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_0\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -0.000235) (not (<= y 1.6e-5)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+
       3.0
       (+ (* 1.5 (* (cos x) t_0)) (* 6.0 (/ 1.0 (+ 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -0.000235) || !(y <= 1.6e-5)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_0)) + (6.0 * (1.0 / (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) + (-1.0d0)
    if ((y <= (-0.000235d0)) .or. (.not. (y <= 1.6d-5))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((1.5d0 * (cos(x) * t_0)) + (6.0d0 * (1.0d0 / (3.0d0 + sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -0.000235) || !(y <= 1.6e-5)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((1.5 * (Math.cos(x) * t_0)) + (6.0 * (1.0 / (3.0 + Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -0.000235) or not (y <= 1.6e-5):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((1.5 * (math.cos(x) * t_0)) + (6.0 * (1.0 / (3.0 + math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -0.000235) || !(y <= 1.6e-5))
		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(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	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(1.5 * Float64(cos(x) * t_0)) + Float64(6.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -0.000235) || ~((y <= 1.6e-5)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_0)) + (6.0 * (1.0 / (3.0 + sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.000235], N[Not[LessEqual[y, 1.6e-5]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.34999999999999993e-4 or 1.59999999999999993e-5 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 59.8%

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

      1. *-commutative 59.8%

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

   5. Simplified 59.8%

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

   if -2.34999999999999993e-4 < y < 1.59999999999999993e-5

   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%

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

      1. flip-- 99.3%

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

      2. metadata-eval 99.3%

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

      3. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. fma-undefine 99.6%

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

      2. associate-*l/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. +-commutative 99.6%

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

       2. associate-*r* 99.6%

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

       3. fma-define 99.6%

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

       4. associate-*r/ 99.6%

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

   12. Applied egg-rr 99.6%

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

   13. Taylor expanded in y around 0 98.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000235 \lor \neg \left(y \leq 1.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00023:\\ \;\;\;\;\frac{t\_1}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_2\right) + 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_0} + 1.5 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \frac{4}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.00023)
     (/ t_1 (+ 3.0 (+ (* 1.5 (* (cos x) t_2)) (* 6.0 (/ 1.0 t_0)))))
     (if (<= x 1.05e-5)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_2))))
       (*
        0.3333333333333333
        (/
         t_1
         (+
          1.0
          (+ (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)) (* 0.5 (/ 4.0 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.00023) {
		tmp = t_1 / (3.0 + ((1.5 * (cos(x) * t_2)) + (6.0 * (1.0 / t_0))));
	} else if (x <= 1.05e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_2)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (4.0 / t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    t_2 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-0.00023d0)) then
        tmp = t_1 / (3.0d0 + ((1.5d0 * (cos(x) * t_2)) + (6.0d0 * (1.0d0 / t_0))))
    else if (x <= 1.05d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / t_0)) + (1.5d0 * t_2)))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)) + (0.5d0 * (4.0d0 / t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double t_2 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.00023) {
		tmp = t_1 / (3.0 + ((1.5 * (Math.cos(x) * t_2)) + (6.0 * (1.0 / t_0))));
	} else if (x <= 1.05e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / t_0)) + (1.5 * t_2)));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (4.0 / t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	t_2 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -0.00023:
		tmp = t_1 / (3.0 + ((1.5 * (math.cos(x) * t_2)) + (6.0 * (1.0 / t_0))))
	elif x <= 1.05e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / t_0)) + (1.5 * t_2)))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (4.0 / t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.00023)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_2)) + Float64(6.0 * Float64(1.0 / t_0)))));
	elseif (x <= 1.05e-5)
		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(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_2))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(0.5 * Float64(4.0 / t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	t_2 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -0.00023)
		tmp = t_1 / (3.0 + ((1.5 * (cos(x) * t_2)) + (6.0 * (1.0 / t_0))));
	elseif (x <= 1.05e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_2)));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (4.0 / t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.00023], N[(t$95$1 / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-5], 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[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-4

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

   8. Simplified 99.2%

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

      1. fma-undefine 99.2%

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

      2. associate-*l/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. +-commutative 99.2%

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

       2. associate-*r* 99.3%

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

       3. fma-define 99.2%

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

       4. associate-*r/ 99.2%

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

   12. Applied egg-rr 99.2%

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

   13. Taylor expanded in y around 0 58.7%

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

   if -2.3000000000000001e-4 < x < 1.04999999999999994e-5

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. associate-*r* 99.7%

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

       3. fma-define 99.7%

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

       4. associate-*r/ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in x around 0 99.3%

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

   if 1.04999999999999994e-5 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 61.1%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 98.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 98.8%

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

      7. metadata-eval 98.8%

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

      8. metadata-eval 98.8%

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

   7. Applied egg-rr 61.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}} + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative 98.8%

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

   9. Simplified 61.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}} + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00023:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00032 \lor \neg \left(x \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.00032) (not (<= x 3.6e-6)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
     (+
      1.0
      (+ (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)) (* 0.5 (- 3.0 (sqrt 5.0)))))))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
    (+
     3.0
     (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 1.5 (+ (sqrt 5.0) -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.00032) || !(x <= 3.6e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (3.0 - sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.00032d0)) .or. (.not. (x <= 3.6d-6))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (1.5d0 * (sqrt(5.0d0) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.00032) || !(x <= 3.6e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (3.0 - Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (1.5 * (Math.sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.00032) or not (x <= 3.6e-6):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (3.0 - math.sqrt(5.0))))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (1.5 * (math.sqrt(5.0) + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.00032) || !(x <= 3.6e-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(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	else
		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(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(sqrt(5.0) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.00032) || ~((x <= 3.6e-6)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + (0.5 * (3.0 - sqrt(5.0))))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.00032], N[Not[LessEqual[x, 3.6e-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[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000026e-4 or 3.59999999999999984e-6 < x

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 59.7%

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

   if -3.20000000000000026e-4 < x < 3.59999999999999984e-6

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. associate-*r* 99.7%

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

       3. fma-define 99.7%

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

       4. associate-*r/ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in x around 0 99.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00032 \lor \neg \left(x \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_1\right) + 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_0} + 1.5 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -4.8e-6) (not (<= x 1.8e-5)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (+ (* 1.5 (* (cos x) t_1)) (* 6.0 (/ 1.0 t_0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -4.8e-6) || !(x <= 1.8e-5)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-4.8d-6)) .or. (.not. (x <= 1.8d-5))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + ((1.5d0 * (cos(x) * t_1)) + (6.0d0 * (1.0d0 / t_0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / t_0)) + (1.5d0 * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -4.8e-6) || !(x <= 1.8e-5)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + ((1.5 * (Math.cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / t_0)) + (1.5 * t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -4.8e-6) or not (x <= 1.8e-5):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + ((1.5 * (math.cos(x) * t_1)) + (6.0 * (1.0 / t_0))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / t_0)) + (1.5 * t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -4.8e-6) || !(x <= 1.8e-5))
		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(1.5 * Float64(cos(x) * t_1)) + Float64(6.0 * Float64(1.0 / t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -4.8e-6) || ~((x <= 1.8e-5)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -4.8e-6], N[Not[LessEqual[x, 1.8e-5]], $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7999999999999998e-6 or 1.80000000000000005e-5 < x

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 98.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.0%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. metadata-eval 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

   8. Simplified 99.0%

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

      1. fma-undefine 99.0%

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

      2. associate-*l/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. *-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. +-commutative 99.0%

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

       2. associate-*r* 99.1%

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

       3. fma-define 99.0%

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

       4. associate-*r/ 99.1%

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

   12. Applied egg-rr 99.1%

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

   13. Taylor expanded in y around 0 59.8%

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

   if -4.7999999999999998e-6 < x < 1.80000000000000005e-5

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. associate-*r* 99.7%

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

       3. fma-define 99.7%

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

       4. associate-*r/ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in x around 0 99.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-6} \lor \neg \left(x \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(t\_0 - 0.5\right) + 2.5\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -9.5e-6) (not (<= x 6e-5)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (- (+ (* (cos x) (- t_0 0.5)) 2.5) t_0)))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 0.5 (+ t_0 (* (cos y) (- 1.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -9.5e-6) || !(x <= 6e-5)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - 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 + (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) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-9.5d-6)) .or. (.not. (x <= 6d-5))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (((cos(x) * (t_0 - 0.5d0)) + 2.5d0) - 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 + (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) * 0.5;
	double tmp;
	if ((x <= -9.5e-6) || !(x <= 6e-5)) {
		tmp = 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) * (t_0 - 0.5)) + 2.5) - 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 + (Math.cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -9.5e-6) or not (x <= 6e-5):
		tmp = 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) * (t_0 - 0.5)) + 2.5) - 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 + (math.cos(y) * (1.5 - t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -9.5e-6) || !(x <= 6e-5))
		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(Float64(cos(x) * Float64(t_0 - 0.5)) + 2.5) - 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(cos(y) * Float64(1.5 - t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -9.5e-6) || ~((x <= 6e-5)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - 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 + (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] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -9.5e-6], N[Not[LessEqual[x, 6e-5]], $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[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + 2.5), $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[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000005e-6 or 6.00000000000000015e-5 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 59.6%

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

   if -9.5000000000000005e-6 < x < 6.00000000000000015e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-6} \lor \neg \left(x \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2.5\right) - \sqrt{5} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00026 \lor \neg \left(x \leq 2.4 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(t\_0 - 0.5\right) + 2.5\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00026) (not (<= x 2.4e-6)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (- (+ (* (cos x) (- t_0 0.5)) 2.5) t_0)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+
       3.0
       (+
        (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
        (* 1.5 (+ (sqrt 5.0) -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00026) || !(x <= 2.4e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.00026d0)) .or. (.not. (x <= 2.4d-6))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (((cos(x) * (t_0 - 0.5d0)) + 2.5d0) - t_0))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (1.5d0 * (sqrt(5.0d0) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00026) || !(x <= 2.4e-6)) {
		tmp = 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) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (1.5 * (Math.sqrt(5.0) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.00026) or not (x <= 2.4e-6):
		tmp = 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) * (t_0 - 0.5)) + 2.5) - t_0))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (1.5 * (math.sqrt(5.0) + -1.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00026) || !(x <= 2.4e-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(Float64(cos(x) * Float64(t_0 - 0.5)) + 2.5) - t_0)));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * Float64(sqrt(5.0) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.00026) || ~((x <= 2.4e-6)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * (sqrt(5.0) + -1.0))));
	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, -0.00026], N[Not[LessEqual[x, 2.4e-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[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999977e-4 or 2.3999999999999999e-6 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 59.6%

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

   if -2.59999999999999977e-4 < x < 2.3999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      5. pow-prod-up 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}} \cdot 1.5, \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 1.5\right)} \]

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

   8. Simplified 99.8%

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

      1. fma-undefine 99.7%

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

      2. associate-*l/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. +-commutative 99.7%

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

       2. associate-*r* 99.7%

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

       3. fma-define 99.7%

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

       4. associate-*r/ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in x around 0 99.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00026 \lor \neg \left(x \leq 2.4 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2.5\right) - \sqrt{5} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(t\_0 - 0.5\right) + 2.5\right) - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
     (- (+ (* (cos x) (- t_0 0.5)) 2.5) t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) * 0.5d0
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (((cos(x) * (t_0 - 0.5d0)) + 2.5d0) - t_0))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	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) * (t_0 - 0.5)) + 2.5) - t_0));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	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) * (t_0 - 0.5)) + 2.5) - t_0))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	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(Float64(cos(x) * Float64(t_0 - 0.5)) + 2.5) - t_0)))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $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[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 59.0%

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

6. Final simplification 59.0%

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

Alternative 22: 40.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(0.5 \cdot \mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (log
  (+
   1.0
   (expm1
    (*
     0.3333333333333333
     (*
      0.5
      (fma
       -0.0625
       (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))
       2.0)))))))

C

double code(double x, double y) {
	return log((1.0 + expm1((0.3333333333333333 * (0.5 * fma(-0.0625, (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))), 2.0))))));
}

Julia

function code(x, y)
	return log(Float64(1.0 + expm1(Float64(0.3333333333333333 * Float64(0.5 * fma(-0.0625, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))), 2.0))))))
end

Wolfram

code[x_, y_] := N[Log[N[(1.0 + N[(Exp[N[(0.3333333333333333 * N[(0.5 * 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] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $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%

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

5. Taylor expanded in y around 0 59.0%

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

6. Taylor expanded in x around 0 37.6%

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

   1. log1p-expm1-u 37.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2}\right)\right)} \]

   2. log1p-undefine 37.6%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2}\right)\right)} \]

   3. div-inv 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \color{blue}{\left(\left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right) \cdot \frac{1}{2}\right)}\right)\right) \]

   4. +-commutative 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2\right)} \cdot \frac{1}{2}\right)\right)\right) \]

   5. fma-define 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\color{blue}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), 2\right)} \cdot \frac{1}{2}\right)\right)\right) \]

   6. sub-neg 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right), 2\right) \cdot \frac{1}{2}\right)\right)\right) \]

   7. metadata-eval 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right), 2\right) \cdot \frac{1}{2}\right)\right)\right) \]

   8. metadata-eval 37.6%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right) \cdot \color{blue}{0.5}\right)\right)\right) \]

8. Applied egg-rr 37.6%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right) \cdot 0.5\right)\right)\right)} \]

9. Final simplification 37.6%

   \[\leadsto \log \left(1 + \mathsf{expm1}\left(0.3333333333333333 \cdot \left(0.5 \cdot \mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 23: 40.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (expm1
  (log1p
   (+
    0.3333333333333333
    (*
     -0.010416666666666666
     (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ (cos x) -1.0))))))))

C

double code(double x, double y) {
	return expm1(log1p((0.3333333333333333 + (-0.010416666666666666 * (sqrt(2.0) * (pow(sin(x), 2.0) * (cos(x) + -1.0)))))));
}

Java

public static double code(double x, double y) {
	return Math.expm1(Math.log1p((0.3333333333333333 + (-0.010416666666666666 * (Math.sqrt(2.0) * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0)))))));
}

Python

def code(x, y):
	return math.expm1(math.log1p((0.3333333333333333 + (-0.010416666666666666 * (math.sqrt(2.0) * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0)))))))

Julia

function code(x, y)
	return expm1(log1p(Float64(0.3333333333333333 + Float64(-0.010416666666666666 * Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))))))
end

Wolfram

code[x_, y_] := N[(Exp[N[Log[1 + N[(0.3333333333333333 + N[(-0.010416666666666666 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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%

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

5. Taylor expanded in y around 0 59.0%

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

6. Taylor expanded in x around 0 37.6%

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

   1. expm1-log1p-u 37.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2}\right)\right)} \]

   2. expm1-undefine 37.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2}\right)} - 1} \]

8. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right) \cdot 0.5\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-define 37.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \left(\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right) \cdot 0.5\right)\right)\right)} \]

   2. *-commutative 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \color{blue}{\left(0.5 \cdot \mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)\right)}\right)\right) \]

   3. associate-*r* 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.3333333333333333 \cdot 0.5\right) \cdot \mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}\right)\right) \]

   4. metadata-eval 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0.16666666666666666} \cdot \mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)\right)\right) \]

   5. fma-undefine 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.16666666666666666 \cdot \color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right) + 2\right)}\right)\right) \]

   6. metadata-eval 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{\left(-1\right)}\right)\right)\right) + 2\right)\right)\right) \]

   7. sub-neg 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right) + 2\right)\right)\right) \]

   8. +-commutative 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.16666666666666666 \cdot \color{blue}{\left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}\right)\right) \]

   9. distribute-lft-in 37.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0.16666666666666666 \cdot 2 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}\right)\right) \]

   10. metadata-eval 37.6%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0.3333333333333333} + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right) \]

10. Simplified 37.6%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)\right)\right)} \]

11. Final simplification 37.6%

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 24: 40.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  0.3333333333333333
  (*
   -0.010416666666666666
   (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ (cos x) -1.0))))))

C

double code(double x, double y) {
	return 0.3333333333333333 + (-0.010416666666666666 * (sqrt(2.0) * (pow(sin(x), 2.0) * (cos(x) + -1.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 + ((-0.010416666666666666d0) * (sqrt(2.0d0) * ((sin(x) ** 2.0d0) * (cos(x) + (-1.0d0)))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 + (-0.010416666666666666 * (Math.sqrt(2.0) * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) + -1.0))));
}

Python

def code(x, y):
	return 0.3333333333333333 + (-0.010416666666666666 * (math.sqrt(2.0) * (math.pow(math.sin(x), 2.0) * (math.cos(x) + -1.0))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 + Float64(-0.010416666666666666 * Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 + (-0.010416666666666666 * (sqrt(2.0) * ((sin(x) ^ 2.0) * (cos(x) + -1.0))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 + N[(-0.010416666666666666 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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%

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

5. Taylor expanded in y around 0 59.0%

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

6. Taylor expanded in x around 0 37.6%

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

7. Taylor expanded in x around inf 37.6%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in 37.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot 2 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)} \]

   2. metadata-eval 37.6%

      \[\leadsto \color{blue}{0.3333333333333333} + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right) \]

   3. sub-neg 37.6%

      \[\leadsto 0.3333333333333333 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)\right) \]

   4. metadata-eval 37.6%

      \[\leadsto 0.3333333333333333 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)\right) \]

   5. associate-*r* 37.6%

      \[\leadsto 0.3333333333333333 + \color{blue}{\left(0.16666666666666666 \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)} \]

   6. metadata-eval 37.6%

      \[\leadsto 0.3333333333333333 + \color{blue}{-0.010416666666666666} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right) \]

   7. *-commutative 37.6%

      \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot {\sin x}^{2}\right)} \]

   8. metadata-eval 37.6%

      \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + \color{blue}{\left(-1\right)}\right)\right) \cdot {\sin x}^{2}\right) \]

   9. sub-neg 37.6%

      \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right) \cdot {\sin x}^{2}\right) \]

   10. associate-*l* 37.6%

       \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot {\sin x}^{2}\right)\right)} \]

   11. sub-neg 37.6%

       \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot {\sin x}^{2}\right)\right) \]

   12. metadata-eval 37.6%

       \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot {\sin x}^{2}\right)\right) \]

9. Simplified 37.6%

   \[\leadsto \color{blue}{0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)} \]

10. Final simplification 37.6%

    \[\leadsto 0.3333333333333333 + -0.010416666666666666 \cdot \left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right)\right) \]
11. Add Preprocessing

Alternative 25: 40.5% 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%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 59.0%

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

6. Taylor expanded in x around 0 37.6%

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

7. Taylor expanded in x around 0 37.5%

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

8. Final simplification 37.5%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))))