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

Percentage Accurate: 99.3% → 99.4%

Time: 45.1s

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

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

FPCore

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

C

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

Julia

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

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in y around inf 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (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(sin(y) * -0.0625)) * 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(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[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. flip-- 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow1/2 99.1%

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

   4. pow1/2 99.1%

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

   5. pow-prod-up 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

4. Applied egg-rr 99.3%

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

   1. +-commutative 99.4%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     2.0
     (*
      (- (cos x) (cos y))
      (*
       (sqrt 2.0)
       (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))))
    (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	return (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    code = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * (Math.sin(y) - (Math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	return (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * (math.sin(y) - (math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

   5. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in x around inf 99.3%

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

   1. pow-base-1 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*r* 99.3%

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

   4. sub-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. metadata-eval 99.3%

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

   8. cancel-sign-sub-inv 99.3%

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

   9. metadata-eval 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 7: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0114 \lor \neg \left(x \leq 0.0142\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot t\_0\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t\_1 - 0.5\right) + \cos y \cdot \left(1.5 - t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0))) (t_1 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0114) (not (<= x 0.0142)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sqrt 2.0) (sin x)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (* (* (sqrt 2.0) (* (- (sin x) (/ (sin y) 16.0)) t_0)) (- 1.0 (cos y))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0114) || !(x <= 0.0142)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * t_0)) * (1.0 - cos(y)))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.0114d0)) .or. (.not. (x <= 0.0142d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * t_0)) * (1.0d0 - cos(y)))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0114) || !(x <= 0.0142)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_0 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * t_0)) * (1.0 - Math.cos(y)))) / (3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.0114) or not (x <= 0.0142):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_0 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * t_0)) * (1.0 - math.cos(y)))) / (3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0114) || !(x <= 0.0142))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_0 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * t_0)) * Float64(1.0 - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.0114) || ~((x <= 0.0142)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * t_0)) * (1.0 - cos(y)))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0114], N[Not[LessEqual[x, 0.0142]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0114 or 0.014200000000000001 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 60.6%

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

   if -0.0114 < x < 0.014200000000000001

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. cos-neg 99.7%

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

      4. distribute-rgt-in 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0114 \lor \neg \left(x \leq 0.0142\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.061 \lor \neg \left(x \leq 0.042\right):\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

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 -0.061) (not (<= x 0.042)))
     (/ (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (sin x))))) t_1)
     (/
      (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (+ x (* (sin y) -0.0625))))))
      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 <= -0.061) || !(x <= 0.042)) {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * sin(x))))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: 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 <= (-0.061d0)) .or. (.not. (x <= 0.042d0))) then
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * sin(x))))) / t_1
    else
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / 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 <= -0.061) || !(x <= 0.042)) {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * Math.sin(x))))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / 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 <= -0.061) or not (x <= 0.042):
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * math.sin(x))))) / t_1
	else:
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / 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 <= -0.061) || !(x <= 0.042))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * sin(x))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / 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 <= -0.061) || ~((x <= 0.042)))
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * sin(x))))) / t_1;
	else
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / 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, -0.061], N[Not[LessEqual[x, 0.042]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $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[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.060999999999999999 or 0.0420000000000000026 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 60.6%

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

   if -0.060999999999999999 < x < 0.0420000000000000026

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. distribute-rgt-out 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.061 \lor \neg \left(x \leq 0.042\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0038 \lor \neg \left(x \leq 0.00021\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{t\_1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot t\_1\right) + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0038) (not (<= x 0.00021)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_1 2.0)))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (+ (sin x) (* (sin y) -0.0625)) (- 1.0 (cos y))))
       2.0)
      (+ 3.0 (+ (* 1.5 (* (cos y) t_1)) (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0038) || !(x <= 0.00021)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (1.0 - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(y) * t_1)) + (1.5 * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0038) || !(x <= 0.00021))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * t_1)) + Float64(1.5 * t_0))));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0038], N[Not[LessEqual[x, 0.00021]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00379999999999999999 or 2.1000000000000001e-4 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 60.6%

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

   if -0.00379999999999999999 < x < 2.1000000000000001e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.2%

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

   6. Taylor expanded in x around 0 99.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0038 \lor \neg \left(x \leq 0.00021\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot t\_0\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -2.2e+21) (not (<= x 2.65e-5)))
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x)))) 2.0)
      (+
       3.0
       (+ (* 1.5 (* (cos x) t_0)) (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (+ (sin x) (* (sin y) -0.0625)) (- 1.0 (cos y))))
       2.0)
      (+ 3.0 (+ (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0)))) (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -2.2e+21) || !(x <= 2.65e-5)) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))), 2.0) / (3.0 + ((1.5 * (cos(x) * t_0)) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (1.0 - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (1.5 * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -2.2e+21) || !(x <= 2.65e-5))
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_0)) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(1.5 * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -2.2e+21], N[Not[LessEqual[x, 2.65e-5]], $MachinePrecision]], N[(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] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2e21 or 2.65e-5 < x

   1. Initial program 98.8%

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

   3. Simplified 99.0%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around inf 99.1%

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

   10. Taylor expanded in y around 0 58.1%

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

       1. *-commutative 58.1%

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

       2. sub-neg 58.1%

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

       3. metadata-eval 58.1%

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

       4. associate-*l* 58.1%

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

       5. *-commutative 58.1%

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

       6. +-commutative 58.1%

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

       7. distribute-lft-in 58.1%

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

       8. metadata-eval 58.1%

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

   12. Simplified 58.1%

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

   if -2.2e21 < x < 2.65e-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 99.7%

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

   5. Taylor expanded in x around 0 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 2.65 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (+
           (* 1.5 (* (cos x) (+ (sqrt 5.0) -1.0)))
           (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))
   (if (or (<= x -2.2e+21) (not (<= x 112000.0)))
     (/
      (fma (sqrt 2.0) (* (pow (sin x) 2.0) (+ 0.0625 (* -0.0625 (cos x)))) 2.0)
      t_0)
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	double tmp;
	if ((x <= -2.2e+21) || !(x <= 112000.0)) {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * (0.0625 + (-0.0625 * cos(x)))), 2.0) / t_0;
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))
	tmp = 0.0
	if ((x <= -2.2e+21) || !(x <= 112000.0))
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))), 2.0) / t_0);
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -2.2e+21], N[Not[LessEqual[x, 112000.0]], $MachinePrecision]], N[(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] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2e21 or 112000 < x

   1. Initial program 98.8%

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

   3. Simplified 99.0%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around inf 99.1%

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

   10. Taylor expanded in y around 0 59.1%

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

       1. *-commutative 59.1%

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

       2. sub-neg 59.1%

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

       3. metadata-eval 59.1%

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

       4. associate-*l* 59.1%

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

       5. *-commutative 59.1%

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

       6. +-commutative 59.1%

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

       7. distribute-lft-in 59.1%

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

       8. metadata-eval 59.1%

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

   12. Simplified 59.1%

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

   if -2.2e21 < x < 112000

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf 99.7%

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

   10. Taylor expanded in x around 0 95.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \cos x + -1\\ t_2 := 3 + \sqrt{5}\\ t_3 := \sqrt{5} + -1\\ t_4 := \cos x \cdot t\_3\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t\_0\right) \cdot t\_1\right)}{1 + \left(t\_4 + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(1.5 \cdot t\_4 + 6 \cdot \frac{\cos y}{t\_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot t\_1\right) \cdot \left(-0.0625 \cdot t\_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_3}{2}\right) + \cos y \cdot \frac{\frac{4}{t\_2}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (+ (cos x) -1.0))
        (t_2 (+ 3.0 (sqrt 5.0)))
        (t_3 (+ (sqrt 5.0) -1.0))
        (t_4 (* (cos x) t_3)))
   (if (<= x -2.2e+21)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (* (sqrt 2.0) t_0) t_1)))
       (+ 1.0 (* (+ t_4 (* (cos y) (- 3.0 (sqrt 5.0)))) 0.5))))
     (if (<= x 112000.0)
       (/
        (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+ 3.0 (+ (* 1.5 t_4) (* 6.0 (/ (cos y) t_2)))))
       (/
        (+ 2.0 (* (* (sqrt 2.0) t_1) (* -0.0625 t_0)))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ t_3 2.0)))
          (* (cos y) (/ (/ 4.0 t_2) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = cos(x) + -1.0;
	double t_2 = 3.0 + sqrt(5.0);
	double t_3 = sqrt(5.0) + -1.0;
	double t_4 = cos(x) * t_3;
	double tmp;
	if (x <= -2.2e+21) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * t_0) * t_1))) / (1.0 + ((t_4 + (cos(y) * (3.0 - sqrt(5.0)))) * 0.5)));
	} else if (x <= 112000.0) {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((1.5 * t_4) + (6.0 * (cos(y) / t_2))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * t_1) * (-0.0625 * t_0))) / (3.0 * ((1.0 + (cos(x) * (t_3 / 2.0))) + (cos(y) * ((4.0 / t_2) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(cos(x) + -1.0)
	t_2 = Float64(3.0 + sqrt(5.0))
	t_3 = Float64(sqrt(5.0) + -1.0)
	t_4 = Float64(cos(x) * t_3)
	tmp = 0.0
	if (x <= -2.2e+21)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * t_0) * t_1))) / Float64(1.0 + Float64(Float64(t_4 + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) * 0.5))));
	elseif (x <= 112000.0)
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * t_4) + Float64(6.0 * Float64(cos(y) / t_2)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_1) * Float64(-0.0625 * t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_3 / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / t_2) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[x, -2.2e+21], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$4 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 112000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * t$95$4), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e21

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

      4. sub-neg 64.2%

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

      5. metadata-eval 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in x around inf 64.4%

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

   8. Simplified 64.4%

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

   if -2.2e21 < x < 112000

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf 99.7%

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

   10. Taylor expanded in x around 0 95.5%

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

   if 112000 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. *-commutative 53.5%

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

      3. *-commutative 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

   5. Simplified 53.5%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   7. Applied egg-rr 53.6%

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

      1. +-commutative 99.1%

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

   9. Simplified 53.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \cos x + -1\\ t_2 := \sqrt{5} + -1\\ t_3 := \cos x \cdot t\_2 + \cos y \cdot \left(3 - \sqrt{5}\right)\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t\_0\right) \cdot t\_1\right)}{1 + t\_3 \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot t\_1\right) \cdot \left(-0.0625 \cdot t\_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_2}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (+ (cos x) -1.0))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (+ (* (cos x) t_2) (* (cos y) (- 3.0 (sqrt 5.0))))))
   (if (<= x -2.2e+21)
     (*
      0.3333333333333333
      (/ (+ 2.0 (* -0.0625 (* (* (sqrt 2.0) t_0) t_1))) (+ 1.0 (* t_3 0.5))))
     (if (<= x 112000.0)
       (/
        (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
        (+ 3.0 (* 1.5 t_3)))
       (/
        (+ 2.0 (* (* (sqrt 2.0) t_1) (* -0.0625 t_0)))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ t_2 2.0)))
          (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = cos(x) + -1.0;
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = (cos(x) * t_2) + (cos(y) * (3.0 - sqrt(5.0)));
	double tmp;
	if (x <= -2.2e+21) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * t_0) * t_1))) / (1.0 + (t_3 * 0.5)));
	} else if (x <= 112000.0) {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + (1.5 * t_3));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * t_1) * (-0.0625 * t_0))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(cos(x) + -1.0)
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(Float64(cos(x) * t_2) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	tmp = 0.0
	if (x <= -2.2e+21)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * t_0) * t_1))) / Float64(1.0 + Float64(t_3 * 0.5))));
	elseif (x <= 112000.0)
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(1.5 * t_3)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_1) * Float64(-0.0625 * t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e+21], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 112000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e21

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

      4. sub-neg 64.2%

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

      5. metadata-eval 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in x around inf 64.4%

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

   8. Simplified 64.4%

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

   if -2.2e21 < x < 112000

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 95.5%

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

       1. add-cbrt-cube 95.4%

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

       2. pow3 95.4%

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

   11. Applied egg-rr 95.4%

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

   12. Taylor expanded in y around inf 95.5%

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

       1. distribute-lft-out 95.5%

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

       2. *-commutative 95.5%

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

       3. sub-neg 95.5%

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

       4. metadata-eval 95.5%

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

       5. *-commutative 95.5%

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

   14. Simplified 95.5%

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

   if 112000 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. *-commutative 53.5%

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

      3. *-commutative 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

   5. Simplified 53.5%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   7. Applied egg-rr 53.6%

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

      1. +-commutative 99.1%

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

   9. Simplified 53.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{5} + -1\\ t_2 := {\sin x}^{2}\\ t_3 := 3 - \sqrt{5}\\ t_4 := 1 + \cos x \cdot \frac{t\_1}{2}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot t\_2\right) \cdot t\_0\right)}{1 + \left(\cos x \cdot t\_1 + \cos y \cdot t\_3\right) \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(t\_4 + \cos y \cdot \frac{t\_3}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot t\_0\right) \cdot \left(-0.0625 \cdot t\_2\right)}{3 \cdot \left(t\_4 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (cos x) -1.0))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (+ 1.0 (* (cos x) (/ t_1 2.0)))))
   (if (<= x -2.2e+21)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (* (sqrt 2.0) t_2) t_0)))
       (+ 1.0 (* (+ (* (cos x) t_1) (* (cos y) t_3)) 0.5))))
     (if (<= x 112000.0)
       (/
        (+
         2.0
         (* -0.0625 (* (- 1.0 (cos y)) (* (sqrt 2.0) (pow (sin y) 2.0)))))
        (* 3.0 (+ t_4 (* (cos y) (/ t_3 2.0)))))
       (/
        (+ 2.0 (* (* (sqrt 2.0) t_0) (* -0.0625 t_2)))
        (* 3.0 (+ t_4 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) + -1.0;
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = 1.0 + (cos(x) * (t_1 / 2.0));
	double tmp;
	if (x <= -2.2e+21) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * t_2) * t_0))) / (1.0 + (((cos(x) * t_1) + (cos(y) * t_3)) * 0.5)));
	} else if (x <= 112000.0) {
		tmp = (2.0 + (-0.0625 * ((1.0 - cos(y)) * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * (t_4 + (cos(y) * (t_3 / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * t_0) * (-0.0625 * t_2))) / (3.0 * (t_4 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos(x) + (-1.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = sin(x) ** 2.0d0
    t_3 = 3.0d0 - sqrt(5.0d0)
    t_4 = 1.0d0 + (cos(x) * (t_1 / 2.0d0))
    if (x <= (-2.2d+21)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * t_2) * t_0))) / (1.0d0 + (((cos(x) * t_1) + (cos(y) * t_3)) * 0.5d0)))
    else if (x <= 112000.0d0) then
        tmp = (2.0d0 + ((-0.0625d0) * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * (t_4 + (cos(y) * (t_3 / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * t_0) * ((-0.0625d0) * t_2))) / (3.0d0 * (t_4 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) + -1.0;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = 3.0 - Math.sqrt(5.0);
	double t_4 = 1.0 + (Math.cos(x) * (t_1 / 2.0));
	double tmp;
	if (x <= -2.2e+21) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((Math.sqrt(2.0) * t_2) * t_0))) / (1.0 + (((Math.cos(x) * t_1) + (Math.cos(y) * t_3)) * 0.5)));
	} else if (x <= 112000.0) {
		tmp = (2.0 + (-0.0625 * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * Math.pow(Math.sin(y), 2.0))))) / (3.0 * (t_4 + (Math.cos(y) * (t_3 / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * t_0) * (-0.0625 * t_2))) / (3.0 * (t_4 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) + -1.0
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = 3.0 - math.sqrt(5.0)
	t_4 = 1.0 + (math.cos(x) * (t_1 / 2.0))
	tmp = 0
	if x <= -2.2e+21:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * t_2) * t_0))) / (1.0 + (((math.cos(x) * t_1) + (math.cos(y) * t_3)) * 0.5)))
	elif x <= 112000.0:
		tmp = (2.0 + (-0.0625 * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * math.pow(math.sin(y), 2.0))))) / (3.0 * (t_4 + (math.cos(y) * (t_3 / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * t_0) * (-0.0625 * t_2))) / (3.0 * (t_4 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) + -1.0)
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0)))
	tmp = 0.0
	if (x <= -2.2e+21)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * t_2) * t_0))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * t_1) + Float64(cos(y) * t_3)) * 0.5))));
	elseif (x <= 112000.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(t_4 + Float64(cos(y) * Float64(t_3 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_0) * Float64(-0.0625 * t_2))) / Float64(3.0 * Float64(t_4 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) + -1.0;
	t_1 = sqrt(5.0) + -1.0;
	t_2 = sin(x) ^ 2.0;
	t_3 = 3.0 - sqrt(5.0);
	t_4 = 1.0 + (cos(x) * (t_1 / 2.0));
	tmp = 0.0;
	if (x <= -2.2e+21)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * t_2) * t_0))) / (1.0 + (((cos(x) * t_1) + (cos(y) * t_3)) * 0.5)));
	elseif (x <= 112000.0)
		tmp = (2.0 + (-0.0625 * ((1.0 - cos(y)) * (sqrt(2.0) * (sin(y) ^ 2.0))))) / (3.0 * (t_4 + (cos(y) * (t_3 / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * t_0) * (-0.0625 * t_2))) / (3.0 * (t_4 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.2e+21], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 112000.0], N[(N[(2.0 + N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e21

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. *-commutative 64.2%

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

      3. *-commutative 64.2%

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

      4. sub-neg 64.2%

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

      5. metadata-eval 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in x around inf 64.4%

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

   8. Simplified 64.4%

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

   if -2.2e21 < x < 112000

   1. Initial program 99.6%

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

      1. add-cube-cbrt 99.6%

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

      2. pow3 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. pow-base-1 95.5%

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

      3. *-lft-identity 95.5%

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

      4. associate-*r* 95.5%

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

   7. Simplified 95.5%

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

   if 112000 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. *-commutative 53.5%

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

      3. *-commutative 53.5%

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

      4. sub-neg 53.5%

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

      5. metadata-eval 53.5%

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

   5. Simplified 53.5%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow1/2 98.6%

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

      4. pow1/2 98.6%

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

      5. pow-prod-up 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   7. Applied egg-rr 53.6%

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

      1. +-commutative 99.1%

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

   9. Simplified 53.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;x \leq 112000:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot t\_1 + \cos y \cdot t\_0\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -2.2e+21) (not (<= x 112000.0)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0))))
       (+ 1.0 (* (+ (* (cos x) t_1) (* (cos y) t_0)) 0.5))))
     (/
      (+ 2.0 (* -0.0625 (* (- 1.0 (cos y)) (* (sqrt 2.0) (pow (sin y) 2.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0))))))))

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 <= -2.2e+21) || !(x <= 112000.0)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)))) / (1.0 + (((cos(x) * t_1) + (cos(y) * t_0)) * 0.5)));
	} else {
		tmp = (2.0 + (-0.0625 * ((1.0 - cos(y)) * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-2.2d+21)) .or. (.not. (x <= 112000.0d0))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (sin(x) ** 2.0d0)) * (cos(x) + (-1.0d0))))) / (1.0d0 + (((cos(x) * t_1) + (cos(y) * t_0)) * 0.5d0)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -2.2e+21) || !(x <= 112000.0)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0)) * (Math.cos(x) + -1.0)))) / (1.0 + (((Math.cos(x) * t_1) + (Math.cos(y) * t_0)) * 0.5)));
	} else {
		tmp = (2.0 + (-0.0625 * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * Math.pow(Math.sin(y), 2.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	}
	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 <= -2.2e+21) or not (x <= 112000.0):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * math.pow(math.sin(x), 2.0)) * (math.cos(x) + -1.0)))) / (1.0 + (((math.cos(x) * t_1) + (math.cos(y) * t_0)) * 0.5)))
	else:
		tmp = (2.0 + (-0.0625 * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * math.pow(math.sin(y), 2.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	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 <= -2.2e+21) || !(x <= 112000.0))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * t_1) + Float64(cos(y) * t_0)) * 0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	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 <= -2.2e+21) || ~((x <= 112000.0)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (sin(x) ^ 2.0)) * (cos(x) + -1.0)))) / (1.0 + (((cos(x) * t_1) + (cos(y) * t_0)) * 0.5)));
	else
		tmp = (2.0 + (-0.0625 * ((1.0 - cos(y)) * (sqrt(2.0) * (sin(y) ^ 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -2.2e+21], N[Not[LessEqual[x, 112000.0]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2e21 or 112000 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 58.9%

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

      1. associate-*r* 58.9%

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

      2. *-commutative 58.9%

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

      3. *-commutative 58.9%

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

      4. sub-neg 58.9%

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

      5. metadata-eval 58.9%

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

   5. Simplified 58.9%

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

   6. Taylor expanded in x around inf 59.0%

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

   8. Simplified 59.0%

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

   if -2.2e21 < x < 112000

   1. Initial program 99.6%

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

      1. add-cube-cbrt 99.6%

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

      2. pow3 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. pow-base-1 95.5%

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

      3. *-lft-identity 95.5%

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

      4. associate-*r* 95.5%

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

   7. Simplified 95.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+21} \lor \neg \left(x \leq 112000\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot t\_0 + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -3.1e-6) (not (<= x 8.5e-6)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0))))
       (+ 1.0 (* (+ (* (cos x) t_0) (* (cos y) (- 3.0 (sqrt 5.0)))) 0.5))))
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -3.1e-6) || !(x <= 8.5e-6)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)))) / (1.0 + (((cos(x) * t_0) + (cos(y) * (3.0 - sqrt(5.0)))) * 0.5)));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -3.1e-6) || !(x <= 8.5e-6))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * t_0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) * 0.5))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -3.1e-6], N[Not[LessEqual[x, 8.5e-6]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-6 or 8.4999999999999999e-6 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 57.1%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

      3. *-commutative 57.1%

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

      4. sub-neg 57.1%

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

      5. metadata-eval 57.1%

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

   5. Simplified 57.1%

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

   6. Taylor expanded in x around inf 57.2%

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

   8. Simplified 57.2%

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

   if -3.1e-6 < x < 8.4999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 99.3%

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

   10. Taylor expanded in x around 0 99.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t\_0 + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -3.1e-6) (not (<= x 1.55)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0))))
       (+ 1.0 (* 0.5 (+ (* (cos x) t_0) (- 3.0 (sqrt 5.0)))))))
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -3.1e-6) || !(x <= 1.55)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)))) / (1.0 + (0.5 * ((cos(x) * t_0) + (3.0 - sqrt(5.0))))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -3.1e-6) || !(x <= 1.55))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -3.1e-6], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-6 or 1.55000000000000004 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. sub-neg 57.4%

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

      5. metadata-eval 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in y around 0 56.2%

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

      1. sub-neg 56.2%

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

      2. metadata-eval 56.2%

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

      3. associate-*r* 56.2%

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

      4. distribute-lft-out 56.2%

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

      5. sub-neg 56.2%

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

      6. metadata-eval 56.2%

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

   8. Simplified 56.2%

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

   if -3.1e-6 < x < 1.55000000000000004

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 98.7%

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

   10. Taylor expanded in x around 0 98.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -1.76 \cdot 10^{-6} \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot t\_1 + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(t\_1 + \cos y \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -1.76e-6) (not (<= x 1.55)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0))))
       (+ 1.0 (* 0.5 (+ (* (cos x) t_1) t_0)))))
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+ 3.0 (* 1.5 (+ 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 <= -1.76e-6) || !(x <= 1.55)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)))) / (1.0 + (0.5 * ((cos(x) * t_1) + t_0))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + (1.5 * (t_1 + (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 <= -1.76e-6) || !(x <= 1.55))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)))) / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_1) + t_0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_1 + Float64(cos(y) * t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.76e-6], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $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[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7600000000000001e-6 or 1.55000000000000004 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. sub-neg 57.4%

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

      5. metadata-eval 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in y around 0 56.2%

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

      1. sub-neg 56.2%

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

      2. metadata-eval 56.2%

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

      3. associate-*r* 56.2%

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

      4. distribute-lft-out 56.2%

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

      5. sub-neg 56.2%

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

      6. metadata-eval 56.2%

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

   8. Simplified 56.2%

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

   if -1.7600000000000001e-6 < x < 1.55000000000000004

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow1/2 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-prod-up 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 98.7%

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

       1. add-cbrt-cube 98.6%

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

       2. pow3 98.6%

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

   11. Applied egg-rr 98.6%

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

   12. Taylor expanded in x around 0 98.7%

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

       1. distribute-lft-out 98.7%

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

       2. *-commutative 98.7%

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

       3. sub-neg 98.7%

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

       4. metadata-eval 98.7%

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

   14. Simplified 98.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-6} \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right)\right)}{1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (* (sqrt 2.0) (pow (sin x) 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 * ((sqrt(2.0) * pow(sin(x), 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) * ((sqrt(2.0d0) * (sin(x) ** 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.sqrt(2.0) * Math.pow(Math.sin(x), 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.sqrt(2.0) * math.pow(math.sin(x), 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(Float64(sqrt(2.0) * (sin(x) ^ 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 * ((sqrt(2.0) * (sin(x) ^ 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[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $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. Add Preprocessing

3. Taylor expanded in y around 0 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. *-commutative 61.1%

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

   4. sub-neg 61.1%

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

   5. metadata-eval 61.1%

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

5. Simplified 61.1%

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

6. Taylor expanded in y around 0 58.4%

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

   1. sub-neg 58.4%

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

   2. metadata-eval 58.4%

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

   3. associate-*r* 58.4%

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

   4. distribute-lft-out 58.4%

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

   5. sub-neg 58.4%

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

   6. metadata-eval 58.4%

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

8. Simplified 58.4%

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

9. Final simplification 58.4%

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

Alternative 20: 43.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. sub-neg 61.1%

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

   4. metadata-eval 61.1%

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

7. Simplified 61.1%

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

8. Taylor expanded in x around 0 43.7%

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

   1. distribute-lft-out 43.7%

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

10. Simplified 43.7%

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

    1. flip-- 99.1%

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

    2. metadata-eval 99.1%

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

    3. pow1/2 99.1%

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

    4. pow1/2 99.1%

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

    5. pow-prod-up 99.4%

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

    6. metadata-eval 99.4%

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

    7. metadata-eval 99.4%

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

    8. metadata-eval 99.4%

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

12. Applied egg-rr 43.7%

    \[\leadsto \frac{0.6666666666666666}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}\right)} \]
13. Step-by-step derivation

    1. +-commutative 99.4%

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

14. Simplified 43.7%

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

15. Final simplification 43.7%

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

Alternative 21: 43.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 0.6666666666666666 / (0.5 + (0.5 * (sqrt(5.0) + (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 = 0.6666666666666666d0 / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. sub-neg 61.1%

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

   4. metadata-eval 61.1%

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

7. Simplified 61.1%

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

8. Taylor expanded in x around 0 43.7%

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

   1. distribute-lft-out 43.7%

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

10. Simplified 43.7%

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

11. Final simplification 43.7%

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

Alternative 22: 41.2% accurate, 1139.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 61.1%

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

   1. associate-*r* 61.1%

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

   2. *-commutative 61.1%

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

   3. sub-neg 61.1%

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

   4. metadata-eval 61.1%

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

7. Simplified 61.1%

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

8. Taylor expanded in x around 0 43.7%

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

   1. distribute-lft-out 43.7%

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

10. Simplified 43.7%

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

11. Taylor expanded in y around 0 41.3%

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

12. Final simplification 41.3%

    \[\leadsto 0.3333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))))