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

Percentage Accurate: 99.3% → 99.4%

Time: 57.3s

Alternatives: 22

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

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (*
     (+ (sin x) (* -0.0625 (sin y)))
     (* (+ (sin y) (* (sin x) -0.0625)) (log (exp (- (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)) * log(exp((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))) * log(exp((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.log(Math.exp((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.log(math.exp((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)) * log(exp(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)) * log(exp((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[Log[N[Exp[N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]], $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%

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

4. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in y around inf 99.4%

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

    1. add-log-exp 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* (sin x) -0.0625))
    (* (+ (sin x) (* -0.0625 (sin y))) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (*
    1.5
    (+
     (* (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(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) - cos(y)))), 2.0) / (3.0 + (1.5 * ((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(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(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[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(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%

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

4. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))))) / (1.0 + (0.5 * ((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 = 0.3333333333333333d0 * ((2.0d0 + ((sqrt(2.0d0) * (cos(x) - cos(y))) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) - (sin(x) * 0.0625d0))))) / (1.0d0 + (0.5d0 * ((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 0.3333333333333333 * ((2.0 + ((Math.sqrt(2.0) * (Math.cos(x) - Math.cos(y))) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) - (Math.sin(x) * 0.0625))))) / (1.0 + (0.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + ((math.sqrt(2.0) * (math.cos(x) - math.cos(y))) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) - (math.sin(x) * 0.0625))))) / (1.0 + (0.5 * ((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(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) - Float64(sin(x) * 0.0625))))) / Float64(1.0 + Float64(0.5 * 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 = 0.3333333333333333 * ((2.0 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) - (sin(x) * 0.0625))))) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0)))))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 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%

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

4. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

6. Simplified 99.4%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow1/2 99.2%

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

   4. pow1/2 99.2%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in y around inf 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.054 \lor \neg \left(x \leq 0.022\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= x -0.054) (not (<= x 0.022)))
     (/ (+ 2.0 (* t_2 (* (* (sqrt 2.0) (sin x)) t_0))) t_1)
     (/
      (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (+ x (* -0.0625 (sin y)))))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.054) || !(x <= 0.022)) {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = cos(x) - cos(y)
    if ((x <= (-0.054d0)) .or. (.not. (x <= 0.022d0))) then
        tmp = (2.0d0 + (t_2 * ((sqrt(2.0d0) * sin(x)) * t_0))) / t_1
    else
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * (x + ((-0.0625d0) * sin(y))))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.054) || !(x <= 0.022)) {
		tmp = (2.0 + (t_2 * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * (x + (-0.0625 * Math.sin(y))))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.054) or not (x <= 0.022):
		tmp = (2.0 + (t_2 * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / t_1
	else:
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * (x + (-0.0625 * math.sin(y))))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.054) || !(x <= 0.022))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.054) || ~((x <= 0.022)))
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	else
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.054], N[Not[LessEqual[x, 0.022]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0539999999999999994 or 0.021999999999999999 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.4%

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

   if -0.0539999999999999994 < x < 0.021999999999999999

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-out 99.4%

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

      5. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.054 \lor \neg \left(x \leq 0.022\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 6: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0027 \lor \neg \left(x \leq 0.0023\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.0027) (not (<= x 0.0023)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (+ (sin x) (* -0.0625 (sin y)))
         (* (+ (sin y) (* (sin x) -0.0625)) (- 1.0 (cos y))))))
      (+
       3.0
       (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* (cos x) t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.0027) || !(x <= 0.0023)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-0.0027d0)) .or. (.not. (x <= 0.0023d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + ((-0.0625d0) * sin(y))) * ((sin(y) + (sin(x) * (-0.0625d0))) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (cos(x) * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.0027) || !(x <= 0.0023)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (Math.cos(x) * t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -0.0027) or not (x <= 0.0023):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * ((math.sin(y) + (math.sin(x) * -0.0625)) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (math.cos(x) * t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.0027) || !(x <= 0.0023))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(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)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -0.0027) || ~((x <= 0.0023)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (1.0 - cos(y)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0027], N[Not[LessEqual[x, 0.0023]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $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 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[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]), $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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0027000000000000001 or 0.0023 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.6%

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

   if -0.0027000000000000001 < x < 0.0023

   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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around inf 99.7%

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

   10. Taylor expanded in x around 0 99.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0027 \lor \neg \left(x \leq 0.0023\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0024 \lor \neg \left(x \leq 0.00175\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.0024) (not (<= x 0.00175)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) (sin x)) t_0))) t_1)
     (/
      (+
       2.0
       (* (* t_0 (* (sqrt 2.0) (+ x (* -0.0625 (sin y))))) (- 1.0 (cos y))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0024) || !(x <= 0.00175)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))) * (1.0 - cos(y)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.0024d0)) .or. (.not. (x <= 0.00175d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * t_0))) / t_1
    else
        tmp = (2.0d0 + ((t_0 * (sqrt(2.0d0) * (x + ((-0.0625d0) * sin(y))))) * (1.0d0 - cos(y)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0024) || !(x <= 0.00175)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (Math.sqrt(2.0) * (x + (-0.0625 * Math.sin(y))))) * (1.0 - Math.cos(y)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.0024) or not (x <= 0.00175):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / t_1
	else:
		tmp = (2.0 + ((t_0 * (math.sqrt(2.0) * (x + (-0.0625 * math.sin(y))))) * (1.0 - math.cos(y)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.0024) || !(x <= 0.00175))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))) * Float64(1.0 - cos(y)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.0024) || ~((x <= 0.00175)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	else
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))) * (1.0 - cos(y)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0024], N[Not[LessEqual[x, 0.00175]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00239999999999999979 or 0.00175000000000000004 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.6%

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

   if -0.00239999999999999979 < x < 0.00175000000000000004

   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. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-out 99.4%

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

      5. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0024 \lor \neg \left(x \leq 0.00175\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_2, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x + -1\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(t_1 - 0.5\right)\right) - t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0048)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
      (+
       3.0
       (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* (cos x) t_0)))))
     (if (<= x 0.0029)
       (/
        (+
         2.0
         (* (* t_2 (* (sqrt 2.0) (+ x (* -0.0625 (sin y))))) (- 1.0 (cos y))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ t_0 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
       (/
        (fma
         t_2
         (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (+ (cos x) -1.0))
         2.0)
        (* 3.0 (- (+ 2.5 (* (cos x) (- t_1 0.5))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0048) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * t_0))));
	} else if (x <= 0.0029) {
		tmp = (2.0 + ((t_2 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(t_2, ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (cos(x) + -1.0)), 2.0) / (3.0 * ((2.5 + (cos(x) * (t_1 - 0.5))) - t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0048)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * t_0)))));
	elseif (x <= 0.0029)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(fma(t_2, Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(cos(x) + -1.0)), 2.0) / Float64(3.0 * Float64(Float64(2.5 + Float64(cos(x) * Float64(t_1 - 0.5))) - t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0048], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0029], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00479999999999999958

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around inf 98.9%

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

   10. Taylor expanded in y around 0 61.5%

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

       1. *-commutative 61.5%

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

       2. sub-neg 61.5%

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

       3. metadata-eval 61.5%

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

       4. associate-*l* 61.5%

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

       5. *-commutative 61.5%

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

       6. distribute-lft-in 61.5%

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

       7. metadata-eval 61.5%

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

   12. Simplified 61.5%

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

   if -0.00479999999999999958 < x < 0.0029

   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. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-out 99.4%

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

      5. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.2%

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

   if 0.0029 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 63.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   5. Taylor expanded in y around 0 63.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}, 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0048:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x + -1\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]

Alternative 9: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -0.0018:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot t_0\right)}\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{2 + \left(t_2 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_2, \sin x \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(t_1 - 0.5\right)\right) - t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (- (sin y) (/ (sin x) 16.0))))
   (if (<= x -0.0018)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
      (+
       3.0
       (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* (cos x) t_0)))))
     (if (<= x 0.0045)
       (/
        (+
         2.0
         (* (* t_2 (* (sqrt 2.0) (+ x (* -0.0625 (sin y))))) (- 1.0 (cos y))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ t_0 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
       (/
        (fma t_2 (* (sin x) (* (sqrt 2.0) (+ (cos x) -1.0))) 2.0)
        (* 3.0 (- (+ 2.5 (* (cos x) (- t_1 0.5))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if (x <= -0.0018) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (cos(x) * t_0))));
	} else if (x <= 0.0045) {
		tmp = (2.0 + ((t_2 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(t_2, (sin(x) * (sqrt(2.0) * (cos(x) + -1.0))), 2.0) / (3.0 * ((2.5 + (cos(x) * (t_1 - 0.5))) - t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if (x <= -0.0018)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(cos(x) * t_0)))));
	elseif (x <= 0.0045)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(fma(t_2, Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 * Float64(Float64(2.5 + Float64(cos(x) * Float64(t_1 - 0.5))) - t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0018], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0045], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0018

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around inf 98.9%

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

   10. Taylor expanded in y around 0 61.5%

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

       1. *-commutative 61.5%

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

       2. sub-neg 61.5%

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

       3. metadata-eval 61.5%

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

       4. associate-*l* 61.5%

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

       5. *-commutative 61.5%

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

       6. distribute-lft-in 61.5%

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

       7. metadata-eval 61.5%

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

   12. Simplified 61.5%

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

   if -0.0018 < x < 0.00449999999999999966

   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. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-out 99.4%

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

      5. metadata-eval 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.2%

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

   if 0.00449999999999999966 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 63.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   5. Taylor expanded in y around 0 63.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0018:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0045:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \sin x \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]

Alternative 10: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{t_2} + t_0\right)}\\ \mathbf{elif}\;x \leq 0.0007:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \frac{4}{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \sin x \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(t_1 - 0.5\right)\right) - t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (+ 3.0 (sqrt 5.0))))
   (if (<= x -1650000.0)
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) t_2)) t_0))))
     (if (<= x 0.0007)
       (/
        (fma (sqrt 2.0) (* (pow (sin y) 2.0) (* -0.0625 (- 1.0 (cos y)))) 2.0)
        (+ 3.0 (* 1.5 (+ t_0 (* (cos y) (/ 4.0 t_2))))))
       (/
        (fma
         (- (sin y) (/ (sin x) 16.0))
         (* (sin x) (* (sqrt 2.0) (+ (cos x) -1.0)))
         2.0)
        (* 3.0 (- (+ 2.5 (* (cos x) (- t_1 0.5))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -1650000.0) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_2)) + t_0)));
	} else if (x <= 0.0007) {
		tmp = fma(sqrt(2.0), (pow(sin(y), 2.0) * (-0.0625 * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * (4.0 / t_2)))));
	} else {
		tmp = fma((sin(y) - (sin(x) / 16.0)), (sin(x) * (sqrt(2.0) * (cos(x) + -1.0))), 2.0) / (3.0 * ((2.5 + (cos(x) * (t_1 - 0.5))) - t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -1650000.0)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / t_2)) + t_0))));
	elseif (x <= 0.0007)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(y) ^ 2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * Float64(4.0 / t_2))))));
	else
		tmp = Float64(fma(Float64(sin(y) - Float64(sin(x) / 16.0)), Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 * Float64(Float64(2.5 + Float64(cos(x) * Float64(t_1 - 0.5))) - t_1)));
	end
	return 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[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1650000.0], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0007], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e6

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around inf 98.9%

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

   10. Taylor expanded in y around 0 62.7%

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

       1. *-commutative 62.7%

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

       2. sub-neg 62.7%

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

       3. metadata-eval 62.7%

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

       4. associate-*l* 62.7%

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

       5. *-commutative 62.7%

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

       6. distribute-lft-in 62.7%

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

       7. metadata-eval 62.7%

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

   12. Simplified 62.7%

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

   if -1.65e6 < x < 6.99999999999999993e-4

   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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 97.4%

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

       1. *-commutative 97.4%

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

       2. associate-*l* 97.4%

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

       3. *-commutative 97.4%

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

   11. Simplified 97.4%

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

   if 6.99999999999999993e-4 < x

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 63.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   5. Taylor expanded in y around 0 63.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \color{blue}{\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0007:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \sin x \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \left(\left(2.5 + \cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right) - \sqrt{5} \cdot 0.5\right)}\\ \end{array} \]

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := {\sin x}^{2}\\ t_2 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_1 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{t_2} + t_0\right)}\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \frac{4}{t_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot t_1\right) \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (+ 3.0 (sqrt 5.0))))
   (if (<= x -1650000.0)
     (/
      (+ 2.0 (* (sqrt 2.0) (* t_1 (+ 0.0625 (* -0.0625 (cos x))))))
      (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) t_2)) t_0))))
     (if (<= x 0.00088)
       (/
        (fma (sqrt 2.0) (* (pow (sin y) 2.0) (* -0.0625 (- 1.0 (cos y)))) 2.0)
        (+ 3.0 (* 1.5 (+ t_0 (* (cos y) (/ 4.0 t_2))))))
       (/
        (fma (sqrt 2.0) (* (* -0.0625 t_1) (+ (cos x) -1.0)) 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 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -1650000.0) {
		tmp = (2.0 + (sqrt(2.0) * (t_1 * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * ((4.0 * (cos(y) / t_2)) + t_0)));
	} else if (x <= 0.00088) {
		tmp = fma(sqrt(2.0), (pow(sin(y), 2.0) * (-0.0625 * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * (4.0 / t_2)))));
	} else {
		tmp = fma(sqrt(2.0), ((-0.0625 * t_1) * (cos(x) + -1.0)), 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(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -1650000.0)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / t_2)) + t_0))));
	elseif (x <= 0.00088)
		tmp = Float64(fma(sqrt(2.0), Float64((sin(y) ^ 2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * Float64(4.0 / t_2))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(-0.0625 * t_1) * Float64(cos(x) + -1.0)), 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[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1650000.0], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00088], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $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 3 regimes

2. if x < -1.65e6

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around inf 98.9%

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

   10. Taylor expanded in y around 0 62.7%

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

       1. *-commutative 62.7%

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

       2. sub-neg 62.7%

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

       3. metadata-eval 62.7%

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

       4. associate-*l* 62.7%

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

       5. *-commutative 62.7%

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

       6. distribute-lft-in 62.7%

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

       7. metadata-eval 62.7%

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

   12. Simplified 62.7%

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

   if -1.65e6 < x < 8.80000000000000031e-4

   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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 97.4%

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

       1. *-commutative 97.4%

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

       2. associate-*l* 97.4%

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

       3. *-commutative 97.4%

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

   11. Simplified 97.4%

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

   if 8.80000000000000031e-4 < x

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in y around 0 63.2%

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

      1. sub-neg 63.2%

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

      2. metadata-eval 63.2%

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

      3. associate-*r* 63.2%

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

   9. Simplified 63.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 12: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + t_0\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_2 \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{t_1}\\ \mathbf{elif}\;x \leq 0.00075:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot t_2\right) \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_1 (+ 3.0 (* 1.5 (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) t_0))))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -1650000.0)
     (/ (+ 2.0 (* (sqrt 2.0) (* t_2 (+ 0.0625 (* -0.0625 (cos x)))))) t_1)
     (if (<= x 0.00075)
       (/
        (+
         2.0
         (* (sqrt 2.0) (* (pow (sin y) 2.0) (* -0.0625 (- 1.0 (cos y))))))
        t_1)
       (/
        (fma (sqrt 2.0) (* (* -0.0625 t_2) (+ (cos x) -1.0)) 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 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = 3.0 + (1.5 * ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + t_0));
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -1650000.0) {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (0.0625 + (-0.0625 * cos(x)))))) / t_1;
	} else if (x <= 0.00075) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(y), 2.0) * (-0.0625 * (1.0 - cos(y)))))) / t_1;
	} else {
		tmp = fma(sqrt(2.0), ((-0.0625 * t_2) * (cos(x) + -1.0)), 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(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(3.0 + Float64(1.5 * Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + t_0)))
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -1650000.0)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / t_1);
	elseif (x <= 0.00075)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(y) ^ 2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))))) / t_1);
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(-0.0625 * t_2) * Float64(cos(x) + -1.0)), 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[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $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] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -1650000.0], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 0.00075], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $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 3 regimes

2. if x < -1.65e6

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around inf 98.9%

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

   10. Taylor expanded in y around 0 62.7%

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

       1. *-commutative 62.7%

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

       2. sub-neg 62.7%

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

       3. metadata-eval 62.7%

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

       4. associate-*l* 62.7%

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

       5. *-commutative 62.7%

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

       6. distribute-lft-in 62.7%

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

       7. metadata-eval 62.7%

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

   12. Simplified 62.7%

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

   if -1.65e6 < x < 7.5000000000000002e-4

   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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around inf 99.7%

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

   10. Taylor expanded in x around 0 97.4%

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

       1. *-commutative 97.4%

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

       2. associate-*l* 97.4%

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

       3. *-commutative 97.4%

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

   12. Simplified 97.4%

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

   if 7.5000000000000002e-4 < x

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in y around 0 63.2%

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

      1. sub-neg 63.2%

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

      2. metadata-eval 63.2%

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

      3. associate-*r* 63.2%

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

   9. Simplified 63.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650000:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.00075:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 13: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{\cos y}{3 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.00038 \lor \neg \left(x \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(t_0 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
   (if (or (<= x -0.00038) (not (<= x 3.1e-6)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
      (+ 3.0 (* 1.5 (+ t_0 (* (cos x) (+ (sqrt 5.0) -1.0))))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ t_0 -1.0))))))))

C

double code(double x, double y) {
	double t_0 = 4.0 * (cos(y) / (3.0 + sqrt(5.0)));
	double tmp;
	if ((x <= -0.00038) || !(x <= 3.1e-6)) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * (t_0 + (cos(x) * (sqrt(5.0) + -1.0)))));
	} 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) + (t_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 = 4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))
    if ((x <= (-0.00038d0)) .or. (.not. (x <= 3.1d-6))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) ** 2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / (3.0d0 + (1.5d0 * (t_0 + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + (t_0 + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)));
	double tmp;
	if ((x <= -0.00038) || !(x <= 3.1e-6)) {
		tmp = (2.0 + (Math.sqrt(2.0) * (Math.pow(Math.sin(x), 2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / (3.0 + (1.5 * (t_0 + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	} 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) + (t_0 + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))
	tmp = 0
	if (x <= -0.00038) or not (x <= 3.1e-6):
		tmp = (2.0 + (math.sqrt(2.0) * (math.pow(math.sin(x), 2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / (3.0 + (1.5 * (t_0 + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	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) + (t_0 + -1.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))
	tmp = 0.0
	if ((x <= -0.00038) || !(x <= 3.1e-6))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	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(t_0 + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 4.0 * (cos(y) / (3.0 + sqrt(5.0)));
	tmp = 0.0;
	if ((x <= -0.00038) || ~((x <= 3.1e-6)))
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) ^ 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (3.0 + (1.5 * (t_0 + (cos(x) * (sqrt(5.0) + -1.0)))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + (t_0 + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.00038], N[Not[LessEqual[x, 3.1e-6]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $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[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-4 or 3.1e-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%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 98.9%

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

      2. metadata-eval 98.9%

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

      3. pow1/2 98.9%

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

      4. pow1/2 98.9%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in y around inf 99.1%

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

   10. Taylor expanded in y around 0 62.3%

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

       1. *-commutative 62.3%

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

       2. sub-neg 62.3%

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

       3. metadata-eval 62.3%

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

       4. associate-*l* 62.3%

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

       5. *-commutative 62.3%

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

       6. distribute-lft-in 62.3%

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

       7. metadata-eval 62.3%

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

   12. Simplified 62.3%

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

   if -3.8000000000000002e-4 < x < 3.1e-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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

       2. associate--l+ 98.4%

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

       3. +-commutative 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00038 \lor \neg \left(x \leq 3.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \end{array} \]

Alternative 14: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\\ \mathbf{if}\;x \leq -1650000 \lor \neg \left(x \leq 0.0008\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \end{array} \end{array} \]

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 (<= x -1650000.0) (not (<= x 0.0008)))
     (/
      (+
       2.0
       (* (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
      t_0)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (pow (sin y) 2.0) (* -0.0625 (- 1.0 (cos y))))))
      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 ((x <= -1650000.0) || !(x <= 0.0008)) {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (pow(sin(y), 2.0) * (-0.0625 * (1.0 - cos(y)))))) / 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 ((x <= (-1650000.0d0)) .or. (.not. (x <= 0.0008d0))) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) ** 2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(y) ** 2.0d0) * ((-0.0625d0) * (1.0d0 - cos(y)))))) / 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 ((x <= -1650000.0) || !(x <= 0.0008)) {
		tmp = (2.0 + (Math.sqrt(2.0) * (Math.pow(Math.sin(x), 2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (Math.pow(Math.sin(y), 2.0) * (-0.0625 * (1.0 - Math.cos(y)))))) / 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 (x <= -1650000.0) or not (x <= 0.0008):
		tmp = (2.0 + (math.sqrt(2.0) * (math.pow(math.sin(x), 2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (math.pow(math.sin(y), 2.0) * (-0.0625 * (1.0 - math.cos(y)))))) / 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 ((x <= -1650000.0) || !(x <= 0.0008))
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64((sin(y) ^ 2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))))) / 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 ((x <= -1650000.0) || ~((x <= 0.0008)))
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) ^ 2.0) * (0.0625 + (-0.0625 * cos(x)))))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(y) ^ 2.0) * (-0.0625 * (1.0 - cos(y)))))) / 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[x, -1650000.0], N[Not[LessEqual[x, 0.0008]], $MachinePrecision]], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e6 or 8.00000000000000038e-4 < 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%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow1/2 98.8%

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

      4. pow1/2 98.8%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in y around inf 99.1%

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

   10. Taylor expanded in y around 0 63.0%

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

       1. *-commutative 63.0%

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

       2. sub-neg 63.0%

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

       3. metadata-eval 63.0%

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

       4. associate-*l* 63.0%

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

       5. *-commutative 63.0%

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

       6. distribute-lft-in 63.0%

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

       7. metadata-eval 63.0%

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

   12. Simplified 63.0%

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

   if -1.65e6 < x < 8.00000000000000038e-4

   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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around inf 99.7%

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

   10. Taylor expanded in x around 0 97.4%

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

       1. *-commutative 97.4%

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

       2. associate-*l* 97.4%

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

       3. *-commutative 97.4%

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

   12. Simplified 97.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650000 \lor \neg \left(x \leq 0.0008\right):\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 15: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(\cos x \cdot t_2 + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{t_0} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t_2, \frac{4}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.00038)
     (/ t_1 (+ 3.0 (* 1.5 (+ (* (cos x) t_2) (* 4.0 (/ 1.0 t_0))))))
     (if (<= x 6.5e-5)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* 4.0 (/ (cos y) t_0)) -1.0)))))
       (/ t_1 (+ 3.0 (* 1.5 (fma (cos x) t_2 (/ 4.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.00038) {
		tmp = t_1 / (3.0 + (1.5 * ((cos(x) * t_2) + (4.0 * (1.0 / t_0)))));
	} else if (x <= 6.5e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / t_0)) + -1.0))));
	} else {
		tmp = t_1 / (3.0 + (1.5 * fma(cos(x), t_2, (4.0 / t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.00038)
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * t_2) + Float64(4.0 * Float64(1.0 / t_0))))));
	elseif (x <= 6.5e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / t_0)) + -1.0)))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, 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[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.00038], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000002e-4

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around 0 60.3%

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

   if -3.8000000000000002e-4 < x < 6.49999999999999943e-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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

       2. associate--l+ 98.4%

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

       3. +-commutative 98.4%

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

   11. Simplified 98.4%

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

   if 6.49999999999999943e-5 < x

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0 63.1%

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

       1. sub-neg 63.1%

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

       2. metadata-eval 63.1%

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

       3. +-commutative 63.1%

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

       4. *-commutative 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 16: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00039:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(\mathsf{fma}\left(\cos x, t_2, 3\right) - \sqrt{5}\right)}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{t_0} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, t_2, \frac{4}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.00039)
     (*
      0.3333333333333333
      (/ t_1 (+ 1.0 (* 0.5 (- (fma (cos x) t_2 3.0) (sqrt 5.0))))))
     (if (<= x 4.5e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* 4.0 (/ (cos y) t_0)) -1.0)))))
       (/ t_1 (+ 3.0 (* 1.5 (fma (cos x) t_2 (/ 4.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.00039) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (fma(cos(x), t_2, 3.0) - sqrt(5.0)))));
	} else if (x <= 4.5e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / t_0)) + -1.0))));
	} else {
		tmp = t_1 / (3.0 + (1.5 * fma(cos(x), t_2, (4.0 / t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.00039)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(fma(cos(x), t_2, 3.0) - sqrt(5.0))))));
	elseif (x <= 4.5e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / t_0)) + -1.0)))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * fma(cos(x), t_2, 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[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.00039], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999993e-4

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 60.3%

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

      1. *-commutative 60.3%

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

      2. sub-neg 60.3%

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

      3. metadata-eval 60.3%

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

   4. Simplified 60.3%

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

      1. *-un-lft-identity 60.3%

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

      2. associate-+r- 60.2%

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

      3. fma-def 60.3%

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

   6. Applied egg-rr 60.3%

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

   if -3.89999999999999993e-4 < x < 4.50000000000000011e-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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

       2. associate--l+ 98.4%

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

       3. +-commutative 98.4%

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

   11. Simplified 98.4%

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

   if 4.50000000000000011e-6 < x

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. distribute-lft-out 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. pow1/2 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-prod-up 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in y around 0 63.1%

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

       1. sub-neg 63.1%

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

       2. metadata-eval 63.1%

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

       3. +-commutative 63.1%

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

       4. *-commutative 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00039:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}\right)}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 17: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.00038 \lor \neg \left(x \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t_1 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(t_1 + \cos y \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.00038) (not (<= x 1.55e-5)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (* 0.5 (+ (* (cos x) t_1) t_0)))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 1.0 (* 0.5 (+ t_1 (* (cos y) 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 tmp;
	if ((x <= -0.00038) || !(x <= 1.55e-5)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((cos(x) * t_1) + t_0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.0 + (0.5 * (t_1 + (cos(y) * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-0.00038d0)) .or. (.not. (x <= 1.55d-5))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + (0.5d0 * ((cos(x) * t_1) + t_0))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (1.0d0 + (0.5d0 * (t_1 + (cos(y) * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.00038) || !(x <= 1.55e-5)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + (0.5 * ((Math.cos(x) * t_1) + t_0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (1.0 + (0.5 * (t_1 + (Math.cos(y) * t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -0.00038) or not (x <= 1.55e-5):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + (0.5 * ((math.cos(x) * t_1) + t_0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (1.0 + (0.5 * (t_1 + (math.cos(y) * 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)
	tmp = 0.0
	if ((x <= -0.00038) || !(x <= 1.55e-5))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_1) + t_0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(cos(y) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -0.00038) || ~((x <= 1.55e-5)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((cos(x) * t_1) + t_0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (1.0 + (0.5 * (t_1 + (cos(y) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.00038], N[Not[LessEqual[x, 1.55e-5]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-4 or 1.55000000000000007e-5 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. sub-neg 61.6%

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

      3. metadata-eval 61.6%

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

   4. Simplified 61.6%

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

   if -3.8000000000000002e-4 < x < 1.55000000000000007e-5

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. distribute-lft-in 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00038 \lor \neg \left(x \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 18: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00062 \lor \neg \left(x \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.00062) (not (<= x 2.2e-6)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
     (+ 1.0 (* 0.5 (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (- 3.0 (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) (+ (* 4.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.00062) || !(x <= 2.2e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.00062d0)) .or. (.not. (x <= 2.2d-6))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + (0.5d0 * ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) + (3.0d0 - 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) + ((4.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.00062) || !(x <= 2.2e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + (0.5 * ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) + (3.0 - Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (1.5 * (Math.sqrt(5.0) + ((4.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.00062) or not (x <= 2.2e-6):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + (0.5 * ((math.cos(x) * (math.sqrt(5.0) + -1.0)) + (3.0 - math.sqrt(5.0))))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (1.5 * (math.sqrt(5.0) + ((4.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.00062) || !(x <= 2.2e-6))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.00062) || ~((x <= 2.2e-6)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (0.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (3.0 - 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) + ((4.0 * (cos(y) / (3.0 + sqrt(5.0)))) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.00062], N[Not[LessEqual[x, 2.2e-6]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-4 or 2.2000000000000001e-6 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. sub-neg 61.6%

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

      3. metadata-eval 61.6%

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

   4. Simplified 61.6%

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

   if -6.2e-4 < x < 2.2000000000000001e-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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

       2. associate--l+ 98.4%

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

       3. +-commutative 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00062 \lor \neg \left(x \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \end{array} \]

Alternative 19: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_2 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\frac{t_1}{3 + 1.5 \cdot \left(t_2 + 4 \cdot \frac{1}{t_0}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{t_0} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + 0.5 \cdot \left(t_2 + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_2 (* (cos x) (+ (sqrt 5.0) -1.0))))
   (if (<= x -0.00038)
     (/ t_1 (+ 3.0 (* 1.5 (+ t_2 (* 4.0 (/ 1.0 t_0))))))
     (if (<= x 2.2e-6)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (* 1.5 (+ (sqrt 5.0) (+ (* 4.0 (/ (cos y) t_0)) -1.0)))))
       (*
        0.3333333333333333
        (/ t_1 (+ 1.0 (* 0.5 (+ t_2 (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_2 = cos(x) * (sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -0.00038) {
		tmp = t_1 / (3.0 + (1.5 * (t_2 + (4.0 * (1.0 / t_0)))));
	} else if (x <= 2.2e-6) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / t_0)) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    t_2 = cos(x) * (sqrt(5.0d0) + (-1.0d0))
    if (x <= (-0.00038d0)) then
        tmp = t_1 / (3.0d0 + (1.5d0 * (t_2 + (4.0d0 * (1.0d0 / t_0)))))
    else if (x <= 2.2d-6) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (1.5d0 * (sqrt(5.0d0) + ((4.0d0 * (cos(y) / t_0)) + (-1.0d0)))))
    else
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + (0.5d0 * (t_2 + (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double t_2 = Math.cos(x) * (Math.sqrt(5.0) + -1.0);
	double tmp;
	if (x <= -0.00038) {
		tmp = t_1 / (3.0 + (1.5 * (t_2 + (4.0 * (1.0 / t_0)))));
	} else if (x <= 2.2e-6) {
		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) + ((4.0 * (Math.cos(y) / t_0)) + -1.0))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	t_2 = math.cos(x) * (math.sqrt(5.0) + -1.0)
	tmp = 0
	if x <= -0.00038:
		tmp = t_1 / (3.0 + (1.5 * (t_2 + (4.0 * (1.0 / t_0)))))
	elif x <= 2.2e-6:
		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) + ((4.0 * (math.cos(y) / t_0)) + -1.0))))
	else:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_2 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	tmp = 0.0
	if (x <= -0.00038)
		tmp = Float64(t_1 / Float64(3.0 + Float64(1.5 * Float64(t_2 + Float64(4.0 * Float64(1.0 / t_0))))));
	elseif (x <= 2.2e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(1.5 * Float64(sqrt(5.0) + Float64(Float64(4.0 * Float64(cos(y) / t_0)) + -1.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(0.5 * Float64(t_2 + Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	t_2 = cos(x) * (sqrt(5.0) + -1.0);
	tmp = 0.0;
	if (x <= -0.00038)
		tmp = t_1 / (3.0 + (1.5 * (t_2 + (4.0 * (1.0 / t_0)))));
	elseif (x <= 2.2e-6)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (1.5 * (sqrt(5.0) + ((4.0 * (cos(y) / t_0)) + -1.0))));
	else
		tmp = 0.3333333333333333 * (t_1 / (1.0 + (0.5 * (t_2 + (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00038], N[(t$95$1 / N[(3.0 + N[(1.5 * N[(t$95$2 + N[(4.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-6], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(0.5 * N[(t$95$2 + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000002e-4

   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 98.8%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. flip-- 98.7%

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

      2. metadata-eval 98.7%

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

      3. pow1/2 98.7%

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

      4. pow1/2 98.7%

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

      5. pow-prod-up 98.9%

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

      6. metadata-eval 98.9%

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

      7. metadata-eval 98.9%

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

      8. metadata-eval 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in y around 0 60.3%

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

   if -3.8000000000000002e-4 < x < 2.2000000000000001e-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%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. *-commutative 98.4%

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

       2. associate--l+ 98.4%

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

       3. +-commutative 98.4%

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

   11. Simplified 98.4%

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

   if 2.2000000000000001e-6 < x

   1. Initial program 99.1%

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

   2. Taylor expanded in y around 0 63.1%

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

      1. *-commutative 63.1%

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

      2. sub-neg 63.1%

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

      3. metadata-eval 63.1%

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

   4. Simplified 63.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + 4 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 1.5 \cdot \left(\sqrt{5} + \left(4 \cdot \frac{\cos y}{3 + \sqrt{5}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 20: 60.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 61.7%

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

   1. *-commutative 61.7%

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

   2. sub-neg 61.7%

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

   3. metadata-eval 61.7%

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

4. Simplified 61.7%

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

5. Final simplification 61.7%

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

Alternative 21: 42.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around 0 53.0%

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

   1. +-commutative 53.0%

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

   2. associate-*r* 53.0%

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

   3. metadata-eval 53.0%

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

   4. distribute-rgt-out 53.0%

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

   5. metadata-eval 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in x around 0 59.6%

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

6. Taylor expanded in x around 0 63.6%

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

7. Taylor expanded in y around 0 44.5%

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

   1. distribute-lft-out 44.5%

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

   2. sub-neg 44.5%

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

   3. metadata-eval 44.5%

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

9. Simplified 44.5%

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

10. Final simplification 44.5%

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

Alternative 22: 40.3% 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. Taylor expanded in y around 0 61.7%

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

   1. *-commutative 61.7%

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

   2. sub-neg 61.7%

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

   3. metadata-eval 61.7%

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

4. Simplified 61.7%

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

5. Taylor expanded in x around 0 41.9%

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

6. Taylor expanded in x around 0 33.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(-0.5 \cdot \left({x}^{4} \cdot \sqrt{2}\right)\right)}}{1 + 0.5 \cdot 2} \]

7. Taylor expanded in x around 0 41.9%

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

8. Final simplification 41.9%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023308 
(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))))))