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

Percentage Accurate: 99.3% → 99.4%

Time: 49.6s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

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

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
   (*
    1.5
    (fma
     (cos x)
     (+ (sqrt 5.0) -1.0)
     (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))))

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 + (1.5 * fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * (4.0 / (3.0 + sqrt(5.0)))))));
}

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 + Float64(1.5 * fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0))))))))
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[(1.5 * N[(N[Cos[x], $MachinePrecision] * 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%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. fma-def 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

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
    (fma
     (+ (sqrt 5.0) -1.0)
     (cos x)
     (/ (* (cos y) 4.0) (+ 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 * fma((sqrt(5.0) + -1.0), cos(x), ((cos(y) * 4.0) / (3.0 + 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(1.5 * fma(Float64(sqrt(5.0) + -1.0), cos(x), Float64(Float64(cos(y) * 4.0) / Float64(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[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 4.0), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. fma-def 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. *-commutative 99.4%

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

    3. fma-def 99.4%

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

    4. sub-neg 99.4%

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

    5. metadata-eval 99.4%

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

    6. associate-*r/ 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

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
    (+
     (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
     (* (cos x) (+ (sqrt 5.0) -1.0)))))))

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 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
}

Fortran

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

Java

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

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 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[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[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. fma-def 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Simplified 99.3%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

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)) (* (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)) + (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))) + (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)) + (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)) + (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(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + 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)) + (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[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. fma-def 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around inf 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.019 \lor \neg \left(x \leq 0.034\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot t_2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -0.019) (not (<= x 0.034)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* t_1 t_2)))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))
     (/
      (+ 2.0 (* t_1 (* t_2 (* (sqrt 2.0) (+ x (* -0.0625 (sin y)))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -0.019) || !(x <= 0.034)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * t_2))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + (t_1 * (t_2 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = cos(x) - cos(y)
    t_2 = sin(y) - (sin(x) / 16.0d0)
    if ((x <= (-0.019d0)) .or. (.not. (x <= 0.034d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * (t_1 * t_2))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
    else
        tmp = (2.0d0 + (t_1 * (t_2 * (sqrt(2.0d0) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if ((x <= -0.019) || !(x <= 0.034)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * (t_1 * t_2))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + (t_1 * (t_2 * (Math.sqrt(2.0) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if (x <= -0.019) or not (x <= 0.034):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * (t_1 * t_2))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	else:
		tmp = (2.0 + (t_1 * (t_2 * (math.sqrt(2.0) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -0.019) || !(x <= 0.034))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(t_1 * t_2))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(t_2 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = cos(x) - cos(y);
	t_2 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if ((x <= -0.019) || ~((x <= 0.034)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * t_2))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	else
		tmp = (2.0 + (t_1 * (t_2 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.019], N[Not[LessEqual[x, 0.034]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$2), $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], N[(N[(2.0 + N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0189999999999999995 or 0.034000000000000002 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.2%

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

      1. *-commutative 63.2%

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

   7. Simplified 63.2%

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

   if -0.0189999999999999995 < x < 0.034000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. metadata-eval 99.2%

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

      3. distribute-rgt-out 99.2%

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

      4. metadata-eval 99.2%

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

   5. Simplified 99.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019 \lor \neg \left(x \leq 0.034\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0044 \lor \neg \left(x \leq 0.00265\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0044) (not (<= x 0.00265)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (+ (sin x) (* -0.0625 (sin y)))
         (* (- 1.0 (cos y)) (+ (sin y) (* x -0.0625))))))
      (+
       3.0
       (*
        1.5
        (+
         (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
         (* (cos x) (+ (sqrt 5.0) -1.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0044) || !(x <= 0.00265)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0044d0)) .or. (.not. (x <= 0.00265d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((1.0d0 - cos(y)) * (sin(y) + (x * (-0.0625d0))))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0044) || !(x <= 0.00265)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((1.0 - Math.cos(y)) * (Math.sin(y) + (x * -0.0625)))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0044) or not (x <= 0.00265):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * ((math.cos(x) - math.cos(y)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((1.0 - math.cos(y)) * (math.sin(y) + (x * -0.0625)))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0044) || !(x <= 0.00265))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(1.0 - cos(y)) * Float64(sin(y) + Float64(x * -0.0625)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0044) || ~((x <= 0.00265)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((1.0 - cos(y)) * (sin(y) + (x * -0.0625)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0044], N[Not[LessEqual[x, 0.00265]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(x * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00440000000000000027 or 0.00265000000000000001 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.2%

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

      1. *-commutative 63.2%

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

   7. Simplified 63.2%

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

   if -0.00440000000000000027 < x < 0.00265000000000000001

   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. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around inf 99.7%

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

   11. Taylor expanded in x around 0 99.2%

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

       1. +-commutative 99.2%

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

       2. associate-*r* 99.2%

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

       3. distribute-rgt-out 99.2%

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

   13. Simplified 99.2%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \left(\sin y + -0.0625 \cdot x\right)\right)}\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0044 \lor \neg \left(x \leq 0.00265\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(1 - \cos y\right) \cdot \left(\sin y + x \cdot -0.0625\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -480.0) (not (<= x 2.8e+15)))
     (/
      (+
       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)))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin x) (* -0.0625 (sin y)))
        (* (+ (sin y) (* (sin x) -0.0625)) (- 1.0 (cos y))))
       2.0)
      (+
       3.0
       (* 1.5 (+ (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -480.0) || !(x <= 2.8e+15)) {
		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))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 2.8e+15))
		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)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + -1.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 2.8e+15]], $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], 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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 2.8e15 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   8. Taylor expanded in y around 0 61.2%

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

   if -480 < x < 2.8e15

   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. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 97.2%

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

      1. distribute-lft-out 97.2%

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

      2. *-commutative 97.2%

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

      3. sub-neg 97.2%

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

      4. metadata-eval 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in x around 0 97.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x + -0.0625 \cdot \sin y\\ t_1 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;y \leq -0.0052 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sin x) (* -0.0625 (sin y))))
        (t_1
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= y -0.0052) (not (<= y 0.21)))
     (/ (+ 2.0 (* (sqrt 2.0) (* t_0 (* (sin y) (- 1.0 (cos y)))))) t_1)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* t_0 (* (+ (cos x) -1.0) (+ y (* (sin x) -0.0625))))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(x) + (-0.0625 * sin(y));
	double t_1 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.0052) || !(y <= 0.21)) {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * (sin(y) * (1.0 - cos(y)))))) / t_1;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625)))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(x) + ((-0.0625d0) * sin(y))
    t_1 = 3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((y <= (-0.0052d0)) .or. (.not. (y <= 0.21d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_0 * (sin(y) * (1.0d0 - cos(y)))))) / t_1
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_0 * ((cos(x) + (-1.0d0)) * (y + (sin(x) * (-0.0625d0))))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) + (-0.0625 * Math.sin(y));
	double t_1 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.0052) || !(y <= 0.21)) {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_0 * (Math.sin(y) * (1.0 - Math.cos(y)))))) / t_1;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_0 * ((Math.cos(x) + -1.0) * (y + (Math.sin(x) * -0.0625)))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) + (-0.0625 * math.sin(y))
	t_1 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (y <= -0.0052) or not (y <= 0.21):
		tmp = (2.0 + (math.sqrt(2.0) * (t_0 * (math.sin(y) * (1.0 - math.cos(y)))))) / t_1
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_0 * ((math.cos(x) + -1.0) * (y + (math.sin(x) * -0.0625)))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) + Float64(-0.0625 * sin(y)))
	t_1 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((y <= -0.0052) || !(y <= 0.21))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(sin(y) * Float64(1.0 - cos(y)))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(Float64(cos(x) + -1.0) * Float64(y + Float64(sin(x) * -0.0625)))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) + (-0.0625 * sin(y));
	t_1 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((y <= -0.0052) || ~((y <= 0.21)))
		tmp = (2.0 + (sqrt(2.0) * (t_0 * (sin(y) * (1.0 - cos(y)))))) / t_1;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((cos(x) + -1.0) * (y + (sin(x) * -0.0625)))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.0052], N[Not[LessEqual[y, 0.21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0051999999999999998 or 0.209999999999999992 < y

   1. Initial program 99.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in x around 0 56.4%

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

   if -0.0051999999999999998 < y < 0.209999999999999992

   1. Initial program 99.5%

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

   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 y around inf 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. fma-def 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in y around 0 98.7%

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

       1. associate-*r* 98.7%

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

       2. distribute-rgt-out 98.7%

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

       3. sub-neg 98.7%

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

       4. metadata-eval 98.7%

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

   13. Simplified 98.7%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\left(\cos x + -1\right) \cdot \left(-0.0625 \cdot \sin x + y\right)\right)}\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0052 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\cos x + -1\right) \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(t_1 \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -480.0) (not (<= x 2.8e+15)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* t_1 (+ (cos x) -1.0))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))
     (/
      (+
       2.0
       (* (- 1.0 (cos y)) (* t_1 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
      (*
       3.0
       (+
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))
        (+ 1.0 (* (+ (sqrt 5.0) -1.0) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -480.0) || !(x <= 2.8e+15)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + ((1.0 - cos(y)) * (t_1 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = sin(y) - (sin(x) / 16.0d0)
    if ((x <= (-480.0d0)) .or. (.not. (x <= 2.8d+15))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * sin(x)) * (t_1 * (cos(x) + (-1.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
    else
        tmp = (2.0d0 + ((1.0d0 - cos(y)) * (t_1 * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0)))))) / (3.0d0 * ((cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)) + (1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) * 0.5d0))))
    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.sin(y) - (Math.sin(x) / 16.0);
	double tmp;
	if ((x <= -480.0) || !(x <= 2.8e+15)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * Math.sin(x)) * (t_1 * (Math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	} else {
		tmp = (2.0 + ((1.0 - Math.cos(y)) * (t_1 * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0)))))) / (3.0 * ((Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)) + (1.0 + ((Math.sqrt(5.0) + -1.0) * 0.5))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.sin(y) - (math.sin(x) / 16.0)
	tmp = 0
	if (x <= -480.0) or not (x <= 2.8e+15):
		tmp = (2.0 + ((math.sqrt(2.0) * math.sin(x)) * (t_1 * (math.cos(x) + -1.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	else:
		tmp = (2.0 + ((1.0 - math.cos(y)) * (t_1 * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0)))))) / (3.0 * ((math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)) + (1.0 + ((math.sqrt(5.0) + -1.0) * 0.5))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 2.8e+15))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(t_1 * 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)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = sin(y) - (sin(x) / 16.0);
	tmp = 0.0;
	if ((x <= -480.0) || ~((x <= 2.8e+15)))
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * (t_1 * (cos(x) + -1.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	else
		tmp = (2.0 + ((1.0 - cos(y)) * (t_1 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) * 0.5))));
	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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 2.8e+15]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * 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], N[(N[(2.0 + N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 2.8e15 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   8. Taylor expanded in y around 0 61.2%

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

   if -480 < x < 2.8e15

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 97.2%

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

   4. Taylor expanded in x around 0 97.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x + -0.0625 \cdot \sin y\\ t_1 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;y \leq -0.00132 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_0 \cdot \left(\left(\sin x \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sin x) (* -0.0625 (sin y))))
        (t_1
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= y -0.00132) (not (<= y 0.21)))
     (/ (+ 2.0 (* (sqrt 2.0) (* t_0 (* (sin y) (- 1.0 (cos y)))))) t_1)
     (/
      (+ 2.0 (* (sqrt 2.0) (* t_0 (* (* (sin x) -0.0625) (+ (cos x) -1.0)))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(x) + (-0.0625 * sin(y));
	double t_1 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00132) || !(y <= 0.21)) {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * (sin(y) * (1.0 - cos(y)))))) / t_1;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((sin(x) * -0.0625) * (cos(x) + -1.0))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(x) + ((-0.0625d0) * sin(y))
    t_1 = 3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((y <= (-0.00132d0)) .or. (.not. (y <= 0.21d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_0 * (sin(y) * (1.0d0 - cos(y)))))) / t_1
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_0 * ((sin(x) * (-0.0625d0)) * (cos(x) + (-1.0d0)))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) + (-0.0625 * Math.sin(y));
	double t_1 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00132) || !(y <= 0.21)) {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_0 * (Math.sin(y) * (1.0 - Math.cos(y)))))) / t_1;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_0 * ((Math.sin(x) * -0.0625) * (Math.cos(x) + -1.0))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) + (-0.0625 * math.sin(y))
	t_1 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (y <= -0.00132) or not (y <= 0.21):
		tmp = (2.0 + (math.sqrt(2.0) * (t_0 * (math.sin(y) * (1.0 - math.cos(y)))))) / t_1
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_0 * ((math.sin(x) * -0.0625) * (math.cos(x) + -1.0))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) + Float64(-0.0625 * sin(y)))
	t_1 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((y <= -0.00132) || !(y <= 0.21))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(sin(y) * Float64(1.0 - cos(y)))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(Float64(sin(x) * -0.0625) * Float64(cos(x) + -1.0))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) + (-0.0625 * sin(y));
	t_1 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((y <= -0.00132) || ~((y <= 0.21)))
		tmp = (2.0 + (sqrt(2.0) * (t_0 * (sin(y) * (1.0 - cos(y)))))) / t_1;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_0 * ((sin(x) * -0.0625) * (cos(x) + -1.0))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00132], N[Not[LessEqual[y, 0.21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00132 or 0.209999999999999992 < y

   1. Initial program 99.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in x around 0 56.4%

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

   if -0.00132 < y < 0.209999999999999992

   1. Initial program 99.5%

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

   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 y around inf 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. fma-def 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in y around 0 97.9%

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

       1. associate-*r* 97.9%

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

       2. sub-neg 97.9%

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

       3. metadata-eval 97.9%

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

   13. Simplified 97.9%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\left(-0.0625 \cdot \sin x\right) \cdot \left(\cos x + -1\right)\right)}\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00132 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin x \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0033 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0033) (not (<= x 2.8e+15)))
     (/
      (+
       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)))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) (- 1.0 (cos y))))))
      (+
       3.0
       (*
        1.5
        (+
         (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
         (* (cos x) (+ (sqrt 5.0) -1.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0033) || !(x <= 2.8e+15)) {
		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))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0033d0)) .or. (.not. (x <= 2.8d+15))) then
        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))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0033) || !(x <= 2.8e+15)) {
		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))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0033) or not (x <= 2.8e+15):
		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))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0033) || !(x <= 2.8e+15))
		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)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0033) || ~((x <= 2.8e+15)))
		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))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0033], N[Not[LessEqual[x, 2.8e+15]], $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], 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[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0033 or 2.8e15 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.7%

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

      1. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in y around 0 61.0%

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

   if -0.0033 < x < 2.8e15

   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. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in x around 0 97.8%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0033 \lor \neg \left(x \leq 2.8 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;y \leq -0.00165 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= y -0.00165) (not (<= y 0.21)))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (* (+ (sin x) (* -0.0625 (sin y))) (* (sin y) (- 1.0 (cos y))))))
      t_0)
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00165) || !(y <= 0.21)) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -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) :: tmp
    t_0 = 3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((y <= (-0.00165d0)) .or. (.not. (y <= 0.21d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) * (1.0d0 - cos(y)))))) / t_0
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00165) || !(y <= 0.21)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) * (1.0 - Math.cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (y <= -0.00165) or not (y <= 0.21):
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) * (1.0 - math.cos(y)))))) / t_0
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((y <= -0.00165) || !(y <= 0.21))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((y <= -0.00165) || ~((y <= 0.21)))
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) * (1.0 - cos(y)))))) / t_0;
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00165], N[Not[LessEqual[y, 0.21]], $MachinePrecision]], 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[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00165 or 0.209999999999999992 < y

   1. Initial program 99.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in x around 0 56.4%

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

   if -0.00165 < y < 0.209999999999999992

   1. Initial program 99.5%

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

   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 y around inf 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. fma-def 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in y around 0 97.9%

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

       1. associate-*r* 97.9%

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

       2. sub-neg 97.9%

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

       3. metadata-eval 97.9%

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

   13. Simplified 97.9%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00165 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := 3 + \sqrt{5}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{-5} \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{t_1} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(t_0 + 4 \cdot \frac{1}{t_1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0))) (t_1 (+ 3.0 (sqrt 5.0))))
   (if (or (<= y -1.95e-5) (not (<= y 0.21)))
     (/
      (+ 2.0 (* (sqrt 2.0) (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0)))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) t_1)) t_0))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (* 1.5 (+ t_0 (* 4.0 (/ 1.0 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = 3.0 + sqrt(5.0);
	double tmp;
	if ((y <= -1.95e-5) || !(y <= 0.21)) {
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_1)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    t_1 = 3.0d0 + sqrt(5.0d0)
    if ((y <= (-1.95d-5)) .or. (.not. (y <= 0.21d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((1.0d0 - cos(y)) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / t_1)) + t_0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * (t_0 + (4.0d0 * (1.0d0 / t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double t_1 = 3.0 + Math.sqrt(5.0);
	double tmp;
	if ((y <= -1.95e-5) || !(y <= 0.21)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / t_1)) + t_0)));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	t_1 = 3.0 + math.sqrt(5.0)
	tmp = 0
	if (y <= -1.95e-5) or not (y <= 0.21):
		tmp = (2.0 + (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / t_1)) + t_0)))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if ((y <= -1.95e-5) || !(y <= 0.21))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / t_1)) + t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(4.0 * Float64(1.0 / t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (sqrt(5.0) + -1.0);
	t_1 = 3.0 + sqrt(5.0);
	tmp = 0.0;
	if ((y <= -1.95e-5) || ~((y <= 0.21)))
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * (sin(y) ^ 2.0))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_1)) + t_0)));
	else
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * (t_0 + (4.0 * (1.0 / t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -1.95e-5], N[Not[LessEqual[y, 0.21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$0), $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[(t$95$0 + N[(4.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.95e-5 or 0.209999999999999992 < y

   1. Initial program 99.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in x around 0 56.0%

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

       1. associate-*r* 54.8%

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

   13. Simplified 56.0%

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

   if -1.95e-5 < y < 0.209999999999999992

   1. Initial program 99.5%

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

   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 y around inf 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. fma-def 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in y around 0 97.9%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-5} \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;y \leq -0.00094 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           1.5
           (+
            (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))
            (* (cos x) (+ (sqrt 5.0) -1.0)))))))
   (if (or (<= y -0.00094) (not (<= y 0.21)))
     (/
      (+ 2.0 (* (sqrt 2.0) (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0)))))
      t_0)
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00094) || !(y <= 0.21)) {
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -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) :: tmp
    t_0 = 3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))
    if ((y <= (-0.00094d0)) .or. (.not. (y <= 0.21d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((1.0d0 - cos(y)) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / t_0
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))));
	double tmp;
	if ((y <= -0.00094) || !(y <= 0.21)) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))
	tmp = 0
	if (y <= -0.00094) or not (y <= 0.21):
		tmp = (2.0 + (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / t_0
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))
	tmp = 0.0
	if ((y <= -0.00094) || !(y <= 0.21))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * (sqrt(5.0) + -1.0))));
	tmp = 0.0;
	if ((y <= -0.00094) || ~((y <= 0.21)))
		tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * (sin(y) ^ 2.0))))) / t_0;
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00094], N[Not[LessEqual[y, 0.21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.39999999999999972e-4 or 0.209999999999999992 < y

   1. Initial program 99.0%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in x around 0 56.0%

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

       1. associate-*r* 54.8%

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

   13. Simplified 56.0%

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

   if -9.39999999999999972e-4 < y < 0.209999999999999992

   1. Initial program 99.5%

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

   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 y around inf 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. fma-def 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

      1. flip-- 99.4%

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

      2. metadata-eval 99.4%

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

      3. pow1/2 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   11. Taylor expanded in y around 0 97.9%

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

       1. associate-*r* 97.9%

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

       2. sub-neg 97.9%

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

       3. metadata-eval 97.9%

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

   13. Simplified 97.9%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00094 \lor \neg \left(y \leq 0.21\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_2 \cdot t_3\right)}{3 + 1.5 \cdot \left(\cos x \cdot t_1 + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(t_1 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot t_2\right) \cdot t_3}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t_1, \frac{4}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (sqrt 2.0) (+ (cos x) -1.0))))
   (if (<= x -480.0)
     (/
      (+ 2.0 (* -0.0625 (* t_2 t_3)))
      (+ 3.0 (* 1.5 (+ (* (cos x) t_1) (* 4.0 (/ 1.0 t_0))))))
     (if (<= x 2.2e-5)
       (/
        (fma (sqrt 2.0) (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0))) 2.0)
        (+ 3.0 (* 1.5 (+ t_1 (* (cos y) (- 3.0 (sqrt 5.0)))))))
       (/
        (+ 2.0 (* (* -0.0625 t_2) t_3))
        (+ 3.0 (* 1.5 (fma (cos x) t_1 (/ 4.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = sqrt(2.0) * (cos(x) + -1.0);
	double tmp;
	if (x <= -480.0) {
		tmp = (2.0 + (-0.0625 * (t_2 * t_3))) / (3.0 + (1.5 * ((cos(x) * t_1) + (4.0 * (1.0 / t_0)))));
	} else if (x <= 2.2e-5) {
		tmp = fma(sqrt(2.0), ((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), 2.0) / (3.0 + (1.5 * (t_1 + (cos(y) * (3.0 - sqrt(5.0))))));
	} else {
		tmp = (2.0 + ((-0.0625 * t_2) * t_3)) / (3.0 + (1.5 * fma(cos(x), t_1, (4.0 / t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	tmp = 0.0
	if (x <= -480.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_3))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_1) + Float64(4.0 * Float64(1.0 / t_0))))));
	elseif (x <= 2.2e-5)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_1 + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * t_2) * t_3)) / Float64(3.0 + Float64(1.5 * fma(cos(x), t_1, Float64(4.0 / 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[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -480.0], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-5], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -480

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.3%

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

      1. distribute-lft-out 99.1%

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

      2. fma-def 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in y around 0 61.4%

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

   if -480 < x < 2.1999999999999999e-5

   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. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around 0 98.2%

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

       1. associate-*r* 98.2%

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

   13. Simplified 98.2%

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

   if 2.1999999999999999e-5 < x

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 98.8%

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

      1. distribute-lft-out 98.9%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow1/2 98.8%

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

      4. pow1/2 98.8%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around inf 99.1%

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

   11. Taylor expanded in y around 0 57.2%

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

       1. associate-*r* 57.2%

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

       2. sub-neg 57.2%

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

       3. metadata-eval 57.2%

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

       4. +-commutative 57.2%

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

       5. *-commutative 57.2%

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

   13. Simplified 57.2%

       \[\leadsto \color{blue}{\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \frac{4}{\sqrt{5} + 3}\right) \cdot 1.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot t_0 + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -480.0) (not (<= x 6.6e-6)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (* 1.5 (+ (* (cos x) t_0) (* 4.0 (/ 1.0 (+ 3.0 (sqrt 5.0))))))))
     (/
      (fma (sqrt 2.0) (* (- 1.0 (cos y)) (* -0.0625 (pow (sin y) 2.0))) 2.0)
      (+ 3.0 (* 1.5 (+ t_0 (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -480.0) || !(x <= 6.6e-6)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((cos(x) * t_0) + (4.0 * (1.0 / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = fma(sqrt(2.0), ((1.0 - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 6.6e-6))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_0) + Float64(4.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 6.6e-6]], $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[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(4.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 6.60000000000000034e-6 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow1/2 98.9%

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

      4. pow1/2 98.9%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in y around 0 59.1%

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

   if -480 < x < 6.60000000000000034e-6

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around 0 98.2%

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

       1. associate-*r* 98.2%

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

   13. Simplified 98.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \frac{\cos y \cdot 4}{t_0}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0))))
   (if (or (<= x -480.0) (not (<= x 2.9e-6)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (+ 3.0 (* 1.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (* 4.0 (/ 1.0 t_0))))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ -1.0 (/ (* (cos y) 4.0) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double tmp;
	if ((x <= -480.0) || !(x <= 2.9e-6)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + ((cos(y) * 4.0) / t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    if ((x <= (-480.0d0)) .or. (.not. (x <= 2.9d-6))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (4.0d0 * (1.0d0 / t_0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((-1.0d0) + ((cos(y) * 4.0d0) / t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double tmp;
	if ((x <= -480.0) || !(x <= 2.9e-6)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + (-1.0 + ((Math.cos(y) * 4.0) / t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	tmp = 0
	if (x <= -480.0) or not (x <= 2.9e-6):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + (-1.0 + ((math.cos(y) * 4.0) / t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 2.9e-6))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(4.0 * Float64(1.0 / t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(-1.0 + Float64(Float64(cos(y) * 4.0) / t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	tmp = 0.0;
	if ((x <= -480.0) || ~((x <= 2.9e-6)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (4.0 * (1.0 / t_0)))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + ((cos(y) * 4.0) / t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 2.9e-6]], $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[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / 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[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(-1.0 + N[(N[(N[Cos[y], $MachinePrecision] * 4.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 2.9000000000000002e-6 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow1/2 98.9%

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

      4. pow1/2 98.9%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in y around 0 59.1%

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

   if -480 < x < 2.9000000000000002e-6

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

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

      4. pow1/2 99.5%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around 0 98.2%

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

       1. *-commutative 98.2%

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

       2. associate--l+ 98.2%

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

       3. associate-*r/ 98.2%

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

       4. +-commutative 98.2%

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

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(\sqrt{5} + \left(\frac{4 \cdot \cos y}{\sqrt{5} + 3} - 1\right)\right) \cdot 1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.9 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \frac{\cos y \cdot 4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 9.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \frac{\cos y \cdot 4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -480.0) (not (<= x 9.5e-7)))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
    (+ 3.0 (* 1.5 (- (+ 3.0 (* (cos x) (+ (sqrt 5.0) -1.0))) (sqrt 5.0)))))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
    (+
     3.0
     (* 1.5 (+ (sqrt 5.0) (+ -1.0 (/ (* (cos y) 4.0) (+ 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -480.0) || !(x <= 9.5e-7)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + ((cos(y) * 4.0) / (3.0 + sqrt(5.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-480.0d0)) .or. (.not. (x <= 9.5d-7))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * ((3.0d0 + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) - sqrt(5.0d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((-1.0d0) + ((cos(y) * 4.0d0) / (3.0d0 + sqrt(5.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -480.0) || !(x <= 9.5e-7)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) - 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 + (1.5 * (Math.sqrt(5.0) + (-1.0 + ((Math.cos(y) * 4.0) / (3.0 + Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -480.0) or not (x <= 9.5e-7):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (math.cos(x) * (math.sqrt(5.0) + -1.0))) - 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 + (1.5 * (math.sqrt(5.0) + (-1.0 + ((math.cos(y) * 4.0) / (3.0 + math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 9.5e-7))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) - 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(1.5 * Float64(sqrt(5.0) + Float64(-1.0 + Float64(Float64(cos(y) * 4.0) / Float64(3.0 + sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -480.0) || ~((x <= 9.5e-7)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + ((cos(y) * 4.0) / (3.0 + sqrt(5.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 9.5e-7]], $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[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(-1.0 + N[(N[(N[Cos[y], $MachinePrecision] * 4.0), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 9.5000000000000001e-7 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 59.1%

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

   if -480 < x < 9.5000000000000001e-7

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

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

      4. pow1/2 99.5%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around 0 98.2%

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

       1. *-commutative 98.2%

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

       2. associate--l+ 98.2%

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

       3. associate-*r/ 98.2%

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

       4. +-commutative 98.2%

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

   12. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(\sqrt{5} + \left(\frac{4 \cdot \cos y}{\sqrt{5} + 3} - 1\right)\right) \cdot 1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 9.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \frac{\cos y \cdot 4}{3 + \sqrt{5}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -480.0) (not (<= x 2.2e-5)))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
    (+ 3.0 (* 1.5 (- (+ 3.0 (* (cos x) (+ (sqrt 5.0) -1.0))) (sqrt 5.0)))))
   (/
    (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
    (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ -1.0 (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -480.0) || !(x <= 2.2e-5)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + (cos(y) * (3.0 - sqrt(5.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-480.0d0)) .or. (.not. (x <= 2.2d-5))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (3.0d0 + (1.5d0 * ((3.0d0 + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) - sqrt(5.0d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((-1.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -480.0) || !(x <= 2.2e-5)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) - 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 + (1.5 * (Math.sqrt(5.0) + (-1.0 + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -480.0) or not (x <= 2.2e-5):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (math.cos(x) * (math.sqrt(5.0) + -1.0))) - 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 + (1.5 * (math.sqrt(5.0) + (-1.0 + (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -480.0) || !(x <= 2.2e-5))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(3.0 + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) - 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(1.5 * Float64(sqrt(5.0) + Float64(-1.0 + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -480.0) || ~((x <= 2.2e-5)))
		tmp = (2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 + (1.5 * ((3.0 + (cos(x) * (sqrt(5.0) + -1.0))) - sqrt(5.0))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (-1.0 + (cos(y) * (3.0 - sqrt(5.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -480.0], N[Not[LessEqual[x, 2.2e-5]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(-1.0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 2.1999999999999999e-5 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 59.1%

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

   if -480 < x < 2.1999999999999999e-5

   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. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. fma-def 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. associate--l+ 98.2%

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

      2. *-commutative 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480 \lor \neg \left(x \leq 2.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} + -1\right)\right) - \sqrt{5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(-1 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 59.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. fma-def 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in y around 0 59.9%

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

9. Final simplification 59.9%

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

Alternative 21: 45.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around 0 58.4%

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

4. Taylor expanded in x around 0 52.8%

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

   1. associate-*r* 52.8%

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

   2. metadata-eval 52.8%

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

   3. distribute-rgt-out 52.8%

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

   4. metadata-eval 52.8%

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

6. Simplified 52.8%

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

   1. expm1-log1p-u 52.6%

      \[\leadsto \frac{2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. expm1-udef 52.6%

      \[\leadsto \frac{2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y + x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)\right)} - 1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* 52.6%

      \[\leadsto \frac{2 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y + x\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}\right)} - 1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. fma-def 52.6%

      \[\leadsto \frac{2 + \left(e^{\mathsf{log1p}\left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(-0.0625, \sin y, x\right)}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)\right)} - 1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. div-inv 52.6%

      \[\leadsto \frac{2 + \left(e^{\mathsf{log1p}\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\left(\sin y - \color{blue}{\sin x \cdot \frac{1}{16}}\right) \cdot \left(1 - \cos y\right)\right)\right)} - 1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. metadata-eval 52.6%

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

8. Applied egg-rr 52.6%

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

9. Taylor expanded in y around 0 43.0%

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

10. Final simplification 43.0%

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

Alternative 22: 40.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	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))))) / 2.0)

Julia

function code(x, y)
	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))))) / 2.0))
end

MATLAB

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

Wolfram

code[x_, y_] := 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] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 59.8%

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

6. Taylor expanded in x around 0 38.5%

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

   1. unpow2 38.5%

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

   2. sin-mult 38.5%

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

8. Applied egg-rr 38.5%

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

   1. div-sub 38.5%

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

   2. +-inverses 38.5%

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

   3. cos-0 38.5%

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

   4. metadata-eval 38.5%

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

   5. count-2 38.5%

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

   6. *-commutative 38.5%

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

10. Simplified 38.5%

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

11. Final simplification 38.5%

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

Alternative 23: 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. Step-by-step derivation

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 59.8%

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

6. Taylor expanded in x around 0 38.5%

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

7. Taylor expanded in x around 0 38.5%

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

8. Final simplification 38.5%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))))))