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

Percentage Accurate: 99.3% → 99.4%

Time: 44.2s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)}^{3}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin x) (* -0.0625 (sin y)))
    (* (+ (sin y) (* (sin x) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (/ (- 3.0 (sqrt 5.0)) 0.6666666666666666)
    (* (cos x) (pow (cbrt (+ (sqrt 11.25) -1.5)) 3.0))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + fma(cos(y), ((3.0 - sqrt(5.0)) / 0.6666666666666666), (cos(x) * pow(cbrt((sqrt(11.25) + -1.5)), 3.0))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 0.6666666666666666), Float64(cos(x) * (cbrt(Float64(sqrt(11.25) + -1.5)) ^ 3.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[(N[Sqrt[11.25], $MachinePrecision] + -1.5), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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 99.4%

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

5. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot 1.5}\right)} \]

   2. associate-*l* 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \left(\left(\sqrt{5} - 1\right) \cdot 1.5\right)}\right)} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\left(1.5 \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

   4. sub-neg 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right)\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right)\right)} \]

   6. distribute-lft-in 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\left(1.5 \cdot \sqrt{5} + 1.5 \cdot -1\right)}\right)} \]

   7. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \left(1.5 \cdot \sqrt{5} + \color{blue}{-1.5}\right)\right)} \]

7. Simplified 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \color{blue}{\cos x \cdot \left(1.5 \cdot \sqrt{5} + -1.5\right)}\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{\left(\left(\sqrt[3]{1.5 \cdot \sqrt{5} + -1.5} \cdot \sqrt[3]{1.5 \cdot \sqrt{5} + -1.5}\right) \cdot \sqrt[3]{1.5 \cdot \sqrt{5} + -1.5}\right)}\right)} \]

   2. pow3 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{{\left(\sqrt[3]{1.5 \cdot \sqrt{5} + -1.5}\right)}^{3}}\right)} \]

   3. fma-def 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right)}}\right)}^{3}\right)} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1.5, \sqrt{5}, -1.5\right)}\right)}^{3}}\right)} \]
10. Step-by-step derivation

    1. fma-udef 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(\sqrt[3]{\color{blue}{1.5 \cdot \sqrt{5} + -1.5}}\right)}^{3}\right)} \]

11. Applied egg-rr 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(\sqrt[3]{\color{blue}{1.5 \cdot \sqrt{5} + -1.5}}\right)}^{3}\right)} \]
12. Step-by-step derivation

    1. expm1-log1p-u 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{1.5 \cdot \sqrt{5} + -1.5}\right)\right)\right)}}^{3}\right)} \]

    2. expm1-udef 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{1.5 \cdot \sqrt{5} + -1.5}\right)} - 1\right)}}^{3}\right)} \]

    3. add-sqr-sqrt 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\sqrt{1.5 \cdot \sqrt{5}} \cdot \sqrt{1.5 \cdot \sqrt{5}}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    4. sqrt-unprod 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\color{blue}{\sqrt{\left(1.5 \cdot \sqrt{5}\right) \cdot \left(1.5 \cdot \sqrt{5}\right)}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    5. swap-sqr 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{\color{blue}{\left(1.5 \cdot 1.5\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    6. metadata-eval 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    7. pow1/2 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{2.25 \cdot \left(\color{blue}{{5}^{0.5}} \cdot \sqrt{5}\right)} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    8. pow1/2 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{2.25 \cdot \left({5}^{0.5} \cdot \color{blue}{{5}^{0.5}}\right)} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    9. pow-prod-up 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{2.25 \cdot \color{blue}{{5}^{\left(0.5 + 0.5\right)}}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    10. metadata-eval 97.9%

        \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{2.25 \cdot {5}^{\color{blue}{1}}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    11. metadata-eval 97.9%

        \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{2.25 \cdot \color{blue}{5}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

    12. metadata-eval 97.9%

        \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{\color{blue}{11.25}} + -1.5}\right)} - 1\right)}^{3}\right)} \]

13. Applied egg-rr 97.9%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)} - 1\right)}}^{3}\right)} \]
14. Step-by-step derivation

    1. expm1-def 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)\right)\right)}}^{3}\right)} \]

    2. expm1-log1p 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)}}^{3}\right)} \]

15. Simplified 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)}}^{3}\right)} \]

16. Final simplification 99.5%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \cos x \cdot {\left(\sqrt[3]{\sqrt{11.25} + -1.5}\right)}^{3}\right)} \]
17. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (*
     (+ (sin x) (* -0.0625 (sin y)))
     (* (+ (sin y) (* (sin x) -0.0625)) (- (cos x) (cos 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) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(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) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (cos(x) - cos(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.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.cos(x) - Math.cos(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.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.cos(x) - math.cos(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(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(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) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(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[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[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 99.4%

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

   1. flip-- 44.3%

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

   2. metadata-eval 44.3%

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

   3. pow1/2 44.3%

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

   4. pow1/2 44.3%

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

   5. pow-prod-up 44.3%

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

   6. metadata-eval 44.3%

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

   7. metadata-eval 44.3%

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

   8. metadata-eval 44.3%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in x around inf 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-6} \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sin x \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot t\_0\right) \cdot 0.5 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -8.2e-6) (not (<= x 2.2e-9)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (*
         (sin x)
         (*
          (sqrt 2.0)
          (* (- (cos x) (cos y)) (- (sin y) (* (sin x) 0.0625))))))
       (+
        1.0
        (+ (* (* (cos x) t_0) 0.5) (* 0.5 (* (cos y) (- 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 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -8.2e-6) || !(x <= 2.2e-9)) {
		tmp = 0.3333333333333333 * ((2.0 + (sin(x) * (sqrt(2.0) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * t_0) * 0.5) + (0.5 * (cos(y) * (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 * 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) + (-1.0d0)
    if ((x <= (-8.2d-6)) .or. (.not. (x <= 2.2d-9))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (sin(x) * (sqrt(2.0d0) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) * 0.0625d0)))))) / (1.0d0 + (((cos(x) * t_0) * 0.5d0) + (0.5d0 * (cos(y) * (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 * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -8.2e-6) || !(x <= 2.2e-9)) {
		tmp = 0.3333333333333333 * ((2.0 + (Math.sin(x) * (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (1.0 + (((Math.cos(x) * t_0) * 0.5) + (0.5 * (Math.cos(y) * (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 * t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -8.2e-6) or not (x <= 2.2e-9):
		tmp = 0.3333333333333333 * ((2.0 + (math.sin(x) * (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (1.0 + (((math.cos(x) * t_0) * 0.5) + (0.5 * (math.cos(y) * (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 * t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -8.2e-6) || !(x <= 2.2e-9))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sin(x) * Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * t_0) * 0.5) + Float64(0.5 * Float64(cos(y) * 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 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -8.2e-6) || ~((x <= 2.2e-9)))
		tmp = 0.3333333333333333 * ((2.0 + (sin(x) * (sqrt(2.0) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * t_0) * 0.5) + (0.5 * (cos(y) * (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 * t_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[x, -8.2e-6], N[Not[LessEqual[x, 2.2e-9]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $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 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999994e-6 or 2.1999999999999998e-9 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 60.9%

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

      1. *-commutative 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in x around inf 60.8%

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

   if -8.1999999999999994e-6 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-6} \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sin x \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sin x \cdot \left(\sqrt{2} \cdot \left(t\_0 \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot t\_2\right) \cdot 0.5 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t\_0 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t\_1 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -7.4e-6)
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* (sin x) (* (sqrt 2.0) (* t_0 (- (sin y) (* (sin x) 0.0625))))))
       (+
        1.0
        (+ (* (* (cos x) t_2) 0.5) (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))
     (if (<= x 2.2e-9)
       (/
        (+
         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 t_2))))
       (/
        (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* t_0 (- (sin y) (/ (sin x) 16.0)))))
        (*
         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 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -7.4e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (sin(x) * (sqrt(2.0) * (t_0 * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * t_2) * 0.5) + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	} else if (x <= 2.2e-9) {
		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 * t_2)));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_0 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 - 0.5))) + (cos(y) * (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) :: t_2
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-7.4d-6)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (sin(x) * (sqrt(2.0d0) * (t_0 * (sin(y) - (sin(x) * 0.0625d0)))))) / (1.0d0 + (((cos(x) * t_2) * 0.5d0) + (0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    else if (x <= 2.2d-9) then
        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 * t_2)))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * (t_0 * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 - 0.5d0))) + (cos(y) * (1.5d0 - t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -7.4e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (Math.sin(x) * (Math.sqrt(2.0) * (t_0 * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (1.0 + (((Math.cos(x) * t_2) * 0.5) + (0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	} else if (x <= 2.2e-9) {
		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 * t_2)));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * (t_0 * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 - 0.5))) + (Math.cos(y) * (1.5 - t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -7.4e-6:
		tmp = 0.3333333333333333 * ((2.0 + (math.sin(x) * (math.sqrt(2.0) * (t_0 * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (1.0 + (((math.cos(x) * t_2) * 0.5) + (0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	elif x <= 2.2e-9:
		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 * t_2)))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * (t_0 * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 - 0.5))) + (math.cos(y) * (1.5 - t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -7.4e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sin(x) * Float64(sqrt(2.0) * Float64(t_0 * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * t_2) * 0.5) + Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	elseif (x <= 2.2e-9)
		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 * t_2))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(t_0 * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sqrt(5.0) / 2.0;
	t_2 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -7.4e-6)
		tmp = 0.3333333333333333 * ((2.0 + (sin(x) * (sqrt(2.0) * (t_0 * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * t_2) * 0.5) + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	elseif (x <= 2.2e-9)
		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 * t_2)));
	else
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_0 * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 - 0.5))) + (cos(y) * (1.5 - t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -7.4e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-9], 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 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 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 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4000000000000003e-6

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 56.3%

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

      1. *-commutative 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in x around inf 56.3%

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

   if -7.4000000000000003e-6 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 2.1999999999999998e-9 < 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* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 65.5%

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

      1. *-commutative 65.5%

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

   7. Simplified 65.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sin x \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t\_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (<= x -2.5e-6)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_1 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 2.2e-9)
       (/
        (+
         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 t_1))))
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (sin x))
          (* (- (sin y) (/ (sin x) 16.0)) (+ (cos x) -1.0))))
        (*
         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 <= -2.5e-6) {
		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 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 2.2e-9) {
		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 * t_1)));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-2.5d-6)) 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 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (x <= 2.2d-9) then
        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 * t_1)))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((sin(y) - (sin(x) / 16.0d0)) * (cos(x) + (-1.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -2.5e-6) {
		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 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (x <= 2.2e-9) {
		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 * t_1)));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -2.5e-6:
		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 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif x <= 2.2e-9:
		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 * t_1)))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -2.5e-6)
		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(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 <= 2.2e-9)
		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 * t_1))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -2.5e-6)
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (x <= 2.2e-9)
		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 * t_1)));
	else
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.5e-6], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $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, 2.2e-9], 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 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000002e-6

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 52.4%

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

   if -2.5000000000000002e-6 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 2.1999999999999998e-9 < 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* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 65.5%

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

      1. *-commutative 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in y around 0 63.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} + -1\\ t_2 := 1 + \cos x \cdot \frac{t\_1}{2}\\ t_3 := {\sin x}^{2}\\ t_4 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t\_3\right)\right)}{3 \cdot \left(t\_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(t\_2 + \cos y \cdot \frac{\frac{4}{t\_4}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot t\_1, \frac{6}{t\_4}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (+ 1.0 (* (cos x) (/ t_1 2.0))))
        (t_3 (pow (sin x) 2.0))
        (t_4 (+ 3.0 (sqrt 5.0))))
   (if (<= x -8.5e+33)
     (/
      (+ 2.0 (* t_0 (* -0.0625 (* (sqrt 2.0) t_3))))
      (* 3.0 (+ t_2 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 510.0)
       (/
        (+ 2.0 (* t_0 (* -0.0625 (* (sqrt 2.0) (pow (sin y) 2.0)))))
        (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 t_4) 2.0)))))
       (/
        (+ 2.0 (* -0.0625 (* t_3 (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (+ 3.0 (fma 1.5 (* (cos x) t_1) (/ 6.0 t_4))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = 1.0 + (cos(x) * (t_1 / 2.0));
	double t_3 = pow(sin(x), 2.0);
	double t_4 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -8.5e+33) {
		tmp = (2.0 + (t_0 * (-0.0625 * (sqrt(2.0) * t_3)))) / (3.0 * (t_2 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 510.0) {
		tmp = (2.0 + (t_0 * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * (t_2 + (cos(y) * ((4.0 / t_4) / 2.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + fma(1.5, (cos(x) * t_1), (6.0 / t_4)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0)))
	t_3 = sin(x) ^ 2.0
	t_4 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -8.5e+33)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(-0.0625 * Float64(sqrt(2.0) * t_3)))) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 510.0)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(4.0 / t_4) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_3 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + fma(1.5, Float64(cos(x) * t_1), Float64(6.0 / t_4))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+33], N[(N[(2.0 + N[(t$95$0 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 510.0], N[(N[(2.0 + N[(t$95$0 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / t$95$4), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(6.0 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.4999999999999998e33

   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 y around 0 53.4%

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

   if -8.4999999999999998e33 < x < 510

   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 97.9%

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

      1. flip-- 66.7%

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

      2. metadata-eval 66.7%

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

      3. pow1/2 66.7%

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

      4. pow1/2 66.7%

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

      5. pow-prod-up 66.7%

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

      6. metadata-eval 66.7%

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

      7. metadata-eval 66.7%

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

      8. metadata-eval 66.7%

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

   5. Applied egg-rr 97.9%

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

   if 510 < 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%

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

      1. flip-- 21.8%

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

      2. metadata-eval 21.8%

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

      3. pow1/2 21.8%

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

      4. pow1/2 21.8%

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

      5. pow-prod-up 21.8%

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

      6. metadata-eval 21.8%

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

      7. metadata-eval 21.8%

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

      8. metadata-eval 21.8%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 62.2%

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

      1. sub-neg 62.2%

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

      2. metadata-eval 62.2%

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

      3. fma-def 62.2%

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

      4. *-commutative 62.2%

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

      5. sub-neg 62.2%

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

      6. metadata-eval 62.2%

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

      7. associate-*r/ 62.2%

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

      8. metadata-eval 62.2%

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

      9. +-commutative 62.2%

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

   9. Simplified 62.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -1.48 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_2} + 1.5 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot t\_0, \frac{6}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (+ 3.0 (sqrt 5.0))))
   (if (<= x -1.48e-5)
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) t_1))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 2.2e-9)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) t_2)) (* 1.5 t_0))))
       (/
        (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (+ (cos x) -1.0)))))
        (+ 3.0 (fma 1.5 (* (cos x) t_0) (/ 6.0 t_2))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -1.48e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * t_1)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 2.2e-9) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_2)) + (1.5 * t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + fma(1.5, (cos(x) * t_0), (6.0 / t_2)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -1.48e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * t_1)))) / 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)))));
	elseif (x <= 2.2e-9)
		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_2)) + Float64(1.5 * t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + fma(1.5, Float64(cos(x) * t_0), Float64(6.0 / t_2))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.48e-5], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $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], If[LessEqual[x, 2.2e-9], 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$2), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(6.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4800000000000001e-5

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 52.4%

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

   if -1.4800000000000001e-5 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 2.1999999999999998e-9 < 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%

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

      1. flip-- 24.0%

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

      2. metadata-eval 24.0%

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

      3. pow1/2 24.0%

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

      4. pow1/2 24.0%

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

      5. pow-prod-up 24.0%

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

      6. metadata-eval 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. metadata-eval 62.7%

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

      3. fma-def 62.7%

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

      4. *-commutative 62.7%

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

      5. sub-neg 62.7%

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

      6. metadata-eval 62.7%

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

      7. associate-*r/ 62.7%

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

      8. metadata-eval 62.7%

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

      9. +-commutative 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.48 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-6} \lor \neg \left(y \leq 0.0058\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot t\_0, \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -2.6e-6) (not (<= y 0.0058)))
     (/
      (+ 2.0 (* (- 1.0 (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin y) 2.0)))))
      (*
       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 (fma 1.5 (* (cos x) t_0) (/ 6.0 (+ 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -2.6e-6) || !(y <= 0.0058)) {
		tmp = (2.0 + ((1.0 - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + fma(1.5, (cos(x) * t_0), (6.0 / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -2.6e-6) || !(y <= 0.0058))
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + fma(1.5, Float64(cos(x) * t_0), Float64(6.0 / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -2.6e-6], N[Not[LessEqual[y, 0.0058]], $MachinePrecision]], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000009e-6 or 0.0058 < 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 58.0%

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

   4. Taylor expanded in x around 0 57.9%

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

   if -2.60000000000000009e-6 < y < 0.0058

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

      1. flip-- 63.4%

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

      2. metadata-eval 63.4%

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

      3. pow1/2 63.4%

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

      4. pow1/2 63.4%

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

      5. pow-prod-up 63.4%

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

      6. metadata-eval 63.4%

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

      7. metadata-eval 63.4%

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

      8. metadata-eval 63.4%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. fma-def 98.9%

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

      4. *-commutative 98.9%

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

      5. sub-neg 98.9%

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

      6. metadata-eval 98.9%

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

      7. associate-*r/ 98.9%

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

      8. metadata-eval 98.9%

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

      9. +-commutative 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-6} \lor \neg \left(y \leq 0.0058\right):\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ t_3 := \cos x \cdot t\_2\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + t\_3\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_0} + 1.5 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + \mathsf{fma}\left(1.5, t\_3, \frac{6}{t\_0}\right)}\\ \end{array} \end{array} \]

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))
        (t_3 (* (cos x) t_2)))
   (if (<= x -8e-6)
     (* 0.3333333333333333 (/ t_1 (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) t_3)))))
     (if (<= x 2.2e-9)
       (/
        (+
         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))))
       (/ t_1 (+ 3.0 (fma 1.5 t_3 (/ 6.0 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 t_3 = cos(x) * t_2;
	double tmp;
	if (x <= -8e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + t_3))));
	} else if (x <= 2.2e-9) {
		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 = t_1 / (3.0 + fma(1.5, t_3, (6.0 / 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)
	t_3 = Float64(cos(x) * t_2)
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + t_3)))));
	elseif (x <= 2.2e-9)
		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(t_1 / Float64(3.0 + fma(1.5, t_3, Float64(6.0 / t_0))));
	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[(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]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-9], 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[(t$95$1 / N[(3.0 + N[(1.5 * t$95$3 + N[(6.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 56.3%

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

      1. *-commutative 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in y around 0 50.9%

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

      1. sub-neg 50.9%

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

      2. metadata-eval 50.9%

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

      3. distribute-lft-out 50.9%

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

   8. Simplified 50.9%

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

   if -7.99999999999999964e-6 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 2.1999999999999998e-9 < 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%

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

      1. flip-- 24.0%

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

      2. metadata-eval 24.0%

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

      3. pow1/2 24.0%

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

      4. pow1/2 24.0%

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

      5. pow-prod-up 24.0%

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

      6. metadata-eval 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. metadata-eval 62.7%

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

      3. fma-def 62.7%

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

      4. *-commutative 62.7%

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

      5. sub-neg 62.7%

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

      6. metadata-eval 62.7%

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

      7. associate-*r/ 62.7%

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

      8. metadata-eval 62.7%

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

      9. +-commutative 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ t_3 := \cos x \cdot t\_2\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_1}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + t\_3\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t\_0} + 1.5 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + \left(1.5 \cdot t\_3 + 6 \cdot \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

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))
        (t_3 (* (cos x) t_2)))
   (if (<= x -2.2e-6)
     (* 0.3333333333333333 (/ t_1 (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) t_3)))))
     (if (<= x 2.2e-9)
       (/
        (+
         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))))
       (/ t_1 (+ 3.0 (+ (* 1.5 t_3) (* 6.0 (/ 1.0 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 t_3 = cos(x) * t_2;
	double tmp;
	if (x <= -2.2e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + t_3))));
	} else if (x <= 2.2e-9) {
		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 = t_1 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / 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) :: t_3
    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)
    t_3 = cos(x) * t_2
    if (x <= (-2.2d-6)) then
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + t_3))))
    else if (x <= 2.2d-9) 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 = t_1 / (3.0d0 + ((1.5d0 * t_3) + (6.0d0 * (1.0d0 / 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 t_3 = Math.cos(x) * t_2;
	double tmp;
	if (x <= -2.2e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - Math.sqrt(5.0)) + t_3))));
	} else if (x <= 2.2e-9) {
		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 = t_1 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / 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
	t_3 = math.cos(x) * t_2
	tmp = 0
	if x <= -2.2e-6:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - math.sqrt(5.0)) + t_3))))
	elif x <= 2.2e-9:
		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 = t_1 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / 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)
	t_3 = Float64(cos(x) * t_2)
	tmp = 0.0
	if (x <= -2.2e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + t_3)))));
	elseif (x <= 2.2e-9)
		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(t_1 / Float64(3.0 + Float64(Float64(1.5 * t_3) + Float64(6.0 * Float64(1.0 / 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;
	t_3 = cos(x) * t_2;
	tmp = 0.0;
	if (x <= -2.2e-6)
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + t_3))));
	elseif (x <= 2.2e-9)
		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 = t_1 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / 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]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, -2.2e-6], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-9], 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[(t$95$1 / N[(3.0 + N[(N[(1.5 * t$95$3), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2000000000000001e-6

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 56.3%

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

      1. *-commutative 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in y around 0 50.9%

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

      1. sub-neg 50.9%

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

      2. metadata-eval 50.9%

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

      3. distribute-lft-out 50.9%

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

   8. Simplified 50.9%

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

   if -2.2000000000000001e-6 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

   if 2.1999999999999998e-9 < 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%

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

      1. flip-- 24.0%

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

      2. metadata-eval 24.0%

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

      3. pow1/2 24.0%

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

      4. pow1/2 24.0%

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

      5. pow-prod-up 24.0%

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

      6. metadata-eval 24.0%

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

      7. metadata-eval 24.0%

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

      8. metadata-eval 24.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 62.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00021) (not (<= x 2.2e-9)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) (+ (sqrt 5.0) -1.0)))))))
     (*
      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 <= -0.00021) || !(x <= 2.2e-9)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * (sqrt(5.0) + -1.0))))));
	} 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 <= (-0.00021d0)) .or. (.not. (x <= 2.2d-9))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))))
    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 <= -0.00021) || !(x <= 2.2e-9)) {
		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 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))))));
	} 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 <= -0.00021) or not (x <= 2.2e-9):
		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 + (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))))
	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 <= -0.00021) || !(x <= 2.2e-9))
		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(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))));
	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 <= -0.00021) || ~((x <= 2.2e-9)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * (sqrt(5.0) + -1.0))))));
	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, -0.00021], N[Not[LessEqual[x, 2.2e-9]], $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[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < -2.1000000000000001e-4 or 2.1999999999999998e-9 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 60.9%

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

      1. *-commutative 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in y around 0 56.7%

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

      1. sub-neg 56.7%

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

      2. metadata-eval 56.7%

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

      3. distribute-lft-out 56.7%

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

   8. Simplified 56.7%

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

   if -2.1000000000000001e-4 < x < 2.1999999999999998e-9

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. fma-def 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. cos-neg 99.8%

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

      6. distribute-lft-in 99.7%

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

   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 99.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00021 \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-5} \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -1.7e-5) (not (<= x 2.2e-9)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) 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 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -1.7e-5) || !(x <= 2.2e-9)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * 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 * 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) + (-1.0d0)
    if ((x <= (-1.7d-5)) .or. (.not. (x <= 2.2d-9))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * 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 * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -1.7e-5) || !(x <= 2.2e-9)) {
		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 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * 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 * t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -1.7e-5) or not (x <= 2.2e-9):
		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 + (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * 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 * t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -1.7e-5) || !(x <= 2.2e-9))
		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(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * 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 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -1.7e-5) || ~((x <= 2.2e-9)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * 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 * t_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[x, -1.7e-5], N[Not[LessEqual[x, 2.2e-9]], $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[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e-5 or 2.1999999999999998e-9 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 60.9%

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

      1. *-commutative 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in y around 0 56.7%

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

      1. sub-neg 56.7%

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

      2. metadata-eval 56.7%

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

      3. distribute-lft-out 56.7%

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

   8. Simplified 56.7%

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

   if -1.7e-5 < x < 2.1999999999999998e-9

   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%

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

      1. flip-- 67.4%

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

      2. metadata-eval 67.4%

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

      3. pow1/2 67.4%

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

      4. pow1/2 67.4%

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

      5. pow-prod-up 67.4%

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

      6. metadata-eval 67.4%

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

      7. metadata-eval 67.4%

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

      8. metadata-eval 67.4%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-5} \lor \neg \left(x \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0 64.0%

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

   1. *-commutative 64.0%

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

5. Simplified 64.0%

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

6. Taylor expanded in y around 0 60.2%

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

   1. sub-neg 60.2%

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

   2. metadata-eval 60.2%

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

   3. distribute-lft-out 60.2%

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

8. Simplified 60.2%

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

9. Final simplification 60.2%

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

Alternative 14: 60.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((2.5 + (cos(x) * (t_0 - 0.5))) - 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) * ((sqrt(2.0d0) * (cos(x) + (-1.0d0))) * (0.5d0 - (cos((2.0d0 * x)) / 2.0d0))))) / ((2.5d0 + (cos(x) * (t_0 - 0.5d0))) - 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.sqrt(2.0) * (Math.cos(x) + -1.0)) * (0.5 - (Math.cos((2.0 * x)) / 2.0))))) / ((2.5 + (Math.cos(x) * (t_0 - 0.5))) - t_0));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * (math.cos(x) + -1.0)) * (0.5 - (math.cos((2.0 * x)) / 2.0))))) / ((2.5 + (math.cos(x) * (t_0 - 0.5))) - 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(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0))))) / Float64(Float64(2.5 + Float64(cos(x) * Float64(t_0 - 0.5))) - t_0)))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((2.5 + (cos(x) * (t_0 - 0.5))) - 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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.4%

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

   3. fma-def 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. cos-neg 99.4%

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

   6. distribute-lft-in 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.2%

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

   1. unpow2 60.2%

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

   2. sin-mult 60.2%

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

7. Applied egg-rr 60.2%

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

   1. div-sub 60.2%

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

   2. +-inverses 60.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

   3. cos-0 60.2%

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

   4. metadata-eval 60.2%

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

   5. count-2 60.2%

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

   6. *-commutative 60.2%

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

9. Simplified 60.2%

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

10. Final simplification 60.2%

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

Alternative 15: 42.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + (0.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0)))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.6666666666666666d0 / (1.0d0 + (0.5d0 * ((sqrt(5.0d0) + (-1.0d0)) + (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0)))))))
end function

Java

public static double code(double x, double y) {
	return 0.6666666666666666 / (1.0 + (0.5 * ((Math.sqrt(5.0) + -1.0) + (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0)))))));
}

Python

def code(x, y):
	return 0.6666666666666666 / (1.0 + (0.5 * ((math.sqrt(5.0) + -1.0) + (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0)))))))

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(1.0 + Float64(0.5 * Float64(Float64(sqrt(5.0) + -1.0) + Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.6666666666666666 / (1.0 + (0.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0)))))));
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(1.0 + N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0 64.0%

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

   1. *-commutative 64.0%

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

5. Simplified 64.0%

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

6. Taylor expanded in x around 0 44.3%

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

   1. distribute-lft-in 44.3%

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

   2. sub-neg 44.3%

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

   3. metadata-eval 44.3%

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

8. Simplified 44.3%

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

   1. flip-- 44.3%

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

   2. metadata-eval 44.3%

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

   3. pow1/2 44.3%

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

   4. pow1/2 44.3%

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

   5. pow-prod-up 44.3%

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

   6. metadata-eval 44.3%

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

   7. metadata-eval 44.3%

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

   8. metadata-eval 44.3%

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

10. Applied egg-rr 44.3%

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

11. Final simplification 44.3%

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

Alternative 16: 42.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0 64.0%

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

   1. *-commutative 64.0%

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

5. Simplified 64.0%

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

6. Taylor expanded in x around 0 44.3%

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

   1. distribute-lft-in 44.3%

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

   2. sub-neg 44.3%

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

   3. metadata-eval 44.3%

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

8. Simplified 44.3%

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

9. Final simplification 44.3%

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

Alternative 17: 40.6% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0 64.0%

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

   1. *-commutative 64.0%

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

5. Simplified 64.0%

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

6. Taylor expanded in x around 0 44.3%

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

   1. distribute-lft-in 44.3%

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

   2. sub-neg 44.3%

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

   3. metadata-eval 44.3%

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

8. Simplified 44.3%

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

9. Taylor expanded in y around 0 42.6%

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

10. Final simplification 42.6%

    \[\leadsto 0.3333333333333333 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))