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

Percentage Accurate: 99.3% → 99.3%

Time: 24.5s

Alternatives: 30

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y)))), 2.0) / Float64(3.0 + Float64(1.5 * fma(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[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 99.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.3

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.3

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

8. Applied egg-rr 99.3%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. distribute-lft-in N/A

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

    8. sub-neg N/A

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

    11. distribute-lft-out N/A

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

    12. associate-*r* N/A

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

11. Simplified 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 99.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.3

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.3

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

8. Applied egg-rr 99.3%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. distribute-lft-in N/A

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

    9. sub-neg N/A

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

    10. associate-*r* N/A

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

    11. *-commutative N/A

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

11. Simplified 99.3%

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

Alternative 3: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.29:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_2 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.29)
     (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 1.0)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (* (sqrt 2.0) (fma -0.0625 (sin x) (sin y)))
          (fma
           (* x x)
           (fma
            (* x x)
            (fma (* x x) -0.001388888888888889 0.041666666666666664)
            -0.5)
           (- 1.0 (cos y))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_2 (* (cos y) 0.5))))))
       (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.29) {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 1.0) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), (1.0 - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_2 * (cos(y) * 0.5)))));
	} else {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.29)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 1.0)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_2 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_2 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.29], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.28999999999999998

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.28999999999999998 < x < 1

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.7

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.7

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       2. associate--l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       4. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       6. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       9. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       11. +-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       12. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       14. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       16. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), \color{blue}{1 - \cos y}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       17. lower-cos.f64 99.3

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \color{blue}{\cos y}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   11. Simplified 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   if 1 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.29:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.29:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_2 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.29)
     (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 1.0)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (sqrt 2.0)
          (*
           (fma (sin x) -0.0625 (sin y))
           (-
            (fma
             (* x x)
             (fma
              (* x x)
              (fma (* x x) -0.001388888888888889 0.041666666666666664)
              -0.5)
             1.0)
            (cos y))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_2 (* (cos y) 0.5))))))
       (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.29) {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 1.0) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * (fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0) - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_2 * (cos(y) * 0.5)))));
	} else {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.29)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 1.0)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0) - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_2 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_2 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.29], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.28999999999999998

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.28999999999999998 < x < 1

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      14. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      16. lower-cos.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \color{blue}{\cos y}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   9. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   if 1 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.29:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_2 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.25)
     (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 0.108)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (* (sqrt 2.0) (fma -0.0625 (sin x) (sin y)))
          (fma
           x
           (* x (fma (* x x) 0.041666666666666664 -0.5))
           (- 1.0 (cos y))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_2 (* (cos y) 0.5))))))
       (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.25) {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 0.108) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma(x, (x * fma((x * x), 0.041666666666666664, -0.5)), (1.0 - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_2 * (cos(y) * 0.5)))));
	} else {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.25)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 0.108)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, -0.5)), Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_2 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_2 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.25], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.108], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.25

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.25 < x < 0.107999999999999999

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.7

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.7

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       2. associate--l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} + \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right), 1 - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       7. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       11. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       13. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), \color{blue}{1 - \cos y}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

       14. lower-cos.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \color{blue}{\cos y}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   11. Simplified 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   if 0.107999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_2 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_1 + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.25)
     (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 0.108)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (sqrt 2.0)
          (*
           (fma (sin x) -0.0625 (sin y))
           (fma
            (* x x)
            (fma (* x x) 0.041666666666666664 -0.5)
            (- 1.0 (cos y)))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_2 (* (cos y) 0.5))))))
       (/ t_0 (* 3.0 (+ t_1 (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.25) {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 0.108) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * fma((x * x), fma((x * x), 0.041666666666666664, -0.5), (1.0 - cos(y))))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_2 * (cos(y) * 0.5)))));
	} else {
		tmp = t_0 / (3.0 * (t_1 + (cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.25)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 0.108)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), Float64(1.0 - cos(y))))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_2 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_2 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.25], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.108], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.25

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.25 < x < 0.107999999999999999

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1 - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{24}, \frac{-1}{2}\right), \color{blue}{1 - \cos y}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      13. lower-cos.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \color{blue}{\cos y}\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   9. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)}\right), 2\right)}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   if 0.107999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_2 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_1 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(t\_2 + \cos y \cdot \frac{t\_1}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (<= x -0.029)
     (/ t_0 (* 3.0 (+ t_2 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 0.095)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (* (sqrt 2.0) (fma -0.0625 (sin x) (sin y)))
          (fma (* x x) -0.5 (- 1.0 (cos y))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_1 (* (cos y) 0.5))))))
       (/ t_0 (* 3.0 (+ t_2 (* (cos y) (/ t_1 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if (x <= -0.029) {
		tmp = t_0 / (3.0 * (t_2 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 0.095) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma((x * x), -0.5, (1.0 - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_1 * (cos(y) * 0.5)))));
	} else {
		tmp = t_0 / (3.0 * (t_2 + (cos(y) * (t_1 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 0.095)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_1 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(t_1 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(t$95$0 / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.095], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$1 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.1

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.0290000000000000015 < x < 0.095000000000000001

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.7

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.7

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 99.7

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

   11. Simplified 99.7%

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

   if 0.095000000000000001 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   5. Simplified 62.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))
          (*
           3.0
           (+
            (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
            (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))))
   (if (<= x -0.029)
     t_0
     (if (<= x 0.095)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (*
          (* (sqrt 2.0) (fma -0.0625 (sin x) (sin y)))
          (fma (* x x) -0.5 (- 1.0 (cos y))))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma
           (cos x)
           (fma (sqrt 5.0) 0.5 -0.5)
           (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 0.095) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma((x * x), -0.5, (1.0 - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * 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(2.0 / Float64(3.0 + sqrt(5.0)))))))
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 0.095)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(x), sin(y))) * fma(Float64(x * x), -0.5, Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $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[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 0.095], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 0.095000000000000001 < x

   1. Initial program 98.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.0

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.1

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

   7. Simplified 62.1%

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

   if -0.0290000000000000015 < x < 0.095000000000000001

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.7

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.7

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 99.7

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

   11. Simplified 99.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right) \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_0 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{if}\;y \leq -0.00355:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (fma (sin y) -0.0625 (sin x))
           (* (- (cos x) (cos y)) (* (sin y) (sqrt 2.0)))
           2.0)
          (+
           3.0
           (*
            3.0
            (fma
             (cos x)
             (fma (sqrt 5.0) 0.5 -0.5)
             (* t_0 (* (cos y) 0.5))))))))
   (if (<= y -0.00355)
     t_1
     (if (<= y 0.0024)
       (/
        (fma
         0.3333333333333333
         (*
          (sqrt 2.0)
          (*
           (fma -0.0625 (sin y) (sin x))
           (* (fma (sin x) -0.0625 (sin y)) (+ (cos x) -1.0))))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_0)) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(fma(sin(y), -0.0625, sin(x)), ((cos(x) - cos(y)) * (sin(y) * sqrt(2.0))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_0 * (cos(y) * 0.5)))));
	double tmp;
	if (y <= -0.00355) {
		tmp = t_1;
	} else if (y <= 0.0024) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * (fma(sin(x), -0.0625, sin(y)) * (cos(x) + -1.0)))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_0)), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) * sqrt(2.0))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_0 * Float64(cos(y) * 0.5))))))
	tmp = 0.0
	if (y <= -0.00355)
		tmp = t_1;
	elseif (y <= 0.0024)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(x) + -1.0)))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_0)), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00355], t$95$1, If[LessEqual[y, 0.0024], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0035500000000000002 or 0.00239999999999999979 < y

   1. Initial program 99.1%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.2

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 63.8

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

   11. Simplified 63.8%

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

   if -0.0035500000000000002 < y < 0.00239999999999999979

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 99.4

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y around inf

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

   10. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(\left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00355:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_0 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{if}\;y \leq -0.0175:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.008:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + \mathsf{fma}\left(0.5, y \cdot y, -1\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(t\_0, \mathsf{fma}\left(y \cdot y, -0.25, 0.5\right), \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (fma (sin y) -0.0625 (sin x))
           (* (- (cos x) (cos y)) (* (sin y) (sqrt 2.0)))
           2.0)
          (+
           3.0
           (*
            3.0
            (fma
             (cos x)
             (fma (sqrt 5.0) 0.5 -0.5)
             (* t_0 (* (cos y) 0.5))))))))
   (if (<= y -0.0175)
     t_1
     (if (<= y 0.008)
       (/
        (+
         2.0
         (*
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
          (+ (cos x) (fma 0.5 (* y y) -1.0))))
        (*
         3.0
         (+
          1.0
          (fma
           t_0
           (fma (* y y) -0.25 0.5)
           (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(fma(sin(y), -0.0625, sin(x)), ((cos(x) - cos(y)) * (sin(y) * sqrt(2.0))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_0 * (cos(y) * 0.5)))));
	double tmp;
	if (y <= -0.0175) {
		tmp = t_1;
	} else if (y <= 0.008) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))) * (cos(x) + fma(0.5, (y * y), -1.0)))) / (3.0 * (1.0 + fma(t_0, fma((y * y), -0.25, 0.5), (cos(x) * fma(0.5, sqrt(5.0), -0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) * sqrt(2.0))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_0 * Float64(cos(y) * 0.5))))))
	tmp = 0.0
	if (y <= -0.0175)
		tmp = t_1;
	elseif (y <= 0.008)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))) * Float64(cos(x) + fma(0.5, Float64(y * y), -1.0)))) / Float64(3.0 * Float64(1.0 + fma(t_0, fma(Float64(y * y), -0.25, 0.5), Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0175], t$95$1, If[LessEqual[y, 0.008], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + N[(0.5 * N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$0 * N[(N[(y * y), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.017500000000000002 or 0.0080000000000000002 < y

   1. Initial program 99.1%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 99.2

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

      12. lift-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 99.2

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-sin.f64 63.8

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

   11. Simplified 63.8%

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

   if -0.017500000000000002 < y < 0.0080000000000000002

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

   5. Simplified 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(y \cdot y, -0.25, 0.5\right), \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) \cdot \cos x\right)\right)}} \]

   6. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + \mathsf{fma}\left(0.5, \color{blue}{y \cdot y}, -1\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(y \cdot y, -0.25, 0.5\right), \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) \cdot \cos x\right)\right)} \]

   8. Simplified 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + \mathsf{fma}\left(0.5, y \cdot y, -1\right)\right)}}{3 \cdot \left(1 + \mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(y \cdot y, -0.25, 0.5\right), \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right) \cdot \cos x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0175:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 0.008:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\cos x + \mathsf{fma}\left(0.5, y \cdot y, -1\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(y \cdot y, -0.25, 0.5\right), \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\cos x - \cos y\right) \cdot \left(\sin y \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos y \cdot t\_0\\ t_2 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_3, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_0 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (cos y) t_0))
        (t_2 (fma (sin y) -0.0625 (sin x)))
        (t_3 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.0029)
     (/
      (fma
       t_2
       (* (sqrt 2.0) (* (fma (sin x) -0.0625 (sin y)) (+ (cos x) -1.0)))
       2.0)
      (fma 1.5 (fma (cos x) t_3 t_1) 3.0))
     (if (<= x 0.0165)
       (/
        (fma
         t_2
         (* (* (sqrt 2.0) (- 1.0 (cos y))) (fma -0.0625 x (sin y)))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_0 (* (cos y) 0.5))))))
       (/
        (fma
         0.3333333333333333
         (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
         0.6666666666666666)
        (fma 0.5 (fma t_3 (cos x) t_1) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(y) * t_0;
	double t_2 = fma(sin(y), -0.0625, sin(x));
	double t_3 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.0029) {
		tmp = fma(t_2, (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * (cos(x) + -1.0))), 2.0) / fma(1.5, fma(cos(x), t_3, t_1), 3.0);
	} else if (x <= 0.0165) {
		tmp = fma(t_2, ((sqrt(2.0) * (1.0 - cos(y))) * fma(-0.0625, x, sin(y))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_0 * (cos(y) * 0.5)))));
	} else {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_3, cos(x), t_1), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(y) * t_0)
	t_2 = fma(sin(y), -0.0625, sin(x))
	t_3 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.0029)
		tmp = Float64(fma(t_2, Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(x) + -1.0))), 2.0) / fma(1.5, fma(cos(x), t_3, t_1), 3.0));
	elseif (x <= 0.0165)
		tmp = Float64(fma(t_2, Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * fma(-0.0625, x, sin(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_0 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_3, cos(x), t_1), 1.0));
	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[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.0029], N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0165], N[(N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0029

   1. Initial program 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 59.4

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

   7. Simplified 59.4%

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

   8. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   10. Simplified 59.4%

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

   if -0.0029 < x < 0.016500000000000001

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 99.2

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

   9. Simplified 99.2%

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

   if 0.016500000000000001 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_1 \cdot \left(\sqrt{2} \cdot t\_2\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_0 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot t\_1\right) \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot t\_0\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -0.0029)
     (/
      (fma 0.3333333333333333 (* t_1 (* (sqrt 2.0) t_2)) 0.6666666666666666)
      (fma
       (cos y)
       (/ 2.0 (+ 3.0 (sqrt 5.0)))
       (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
     (if (<= x 0.0165)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (sqrt 2.0) (- 1.0 (cos y))) (fma -0.0625 x (sin y)))
         2.0)
        (+
         3.0
         (*
          3.0
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) (* t_0 (* (cos y) 0.5))))))
       (/
        (fma 0.3333333333333333 (* (* (sqrt 2.0) t_1) t_2) 0.6666666666666666)
        (fma 0.5 (fma (+ (sqrt 5.0) -1.0) (cos x) (* (cos y) t_0)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -0.0029) {
		tmp = fma(0.3333333333333333, (t_1 * (sqrt(2.0) * t_2)), 0.6666666666666666) / fma(cos(y), (2.0 / (3.0 + sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else if (x <= 0.0165) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * (1.0 - cos(y))) * fma(-0.0625, x, sin(y))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_0 * (cos(y) * 0.5)))));
	} else {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * t_1) * t_2), 0.6666666666666666) / fma(0.5, fma((sqrt(5.0) + -1.0), cos(x), (cos(y) * t_0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -0.0029)
		tmp = Float64(fma(0.3333333333333333, Float64(t_1 * Float64(sqrt(2.0) * t_2)), 0.6666666666666666) / fma(cos(y), Float64(2.0 / Float64(3.0 + sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	elseif (x <= 0.0165)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * fma(-0.0625, x, sin(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_0 * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * t_1) * t_2), 0.6666666666666666) / fma(0.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), Float64(cos(y) * t_0)), 1.0));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -0.0029], N[(N[(0.3333333333333333 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0165], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0029

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.4%

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

      1. lift-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{\color{blue}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

      11. lower-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \frac{\cos y \cdot \left(9 - {\left(\sqrt{5}\right)}^{2}\right)}{3 + \sqrt{5}}\right)}} \]

   10. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -0.0029 < x < 0.016500000000000001

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 99.2

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

   9. Simplified 99.2%

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

   if 0.016500000000000001 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.00355:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           3.0
           (fma
            (cos x)
            (fma (sqrt 5.0) 0.5 -0.5)
            (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2 (/ (fma t_1 (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y)))) 2.0) t_0)))
   (if (<= y -0.00355)
     t_2
     (if (<= y 0.0024)
       (/
        (fma
         t_1
         (* (* (sqrt 2.0) (+ (cos x) -1.0)) (fma -0.0625 (sin x) y))
         2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), ((3.0 - sqrt(5.0)) * (cos(y) * 0.5))));
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = fma(t_1, (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 2.0) / t_0;
	double tmp;
	if (y <= -0.00355) {
		tmp = t_2;
	} else if (y <= 0.0024) {
		tmp = fma(t_1, ((sqrt(2.0) * (cos(x) + -1.0)) * fma(-0.0625, sin(x), y)), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5)))))
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(fma(t_1, Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.00355)
		tmp = t_2;
	elseif (y <= 0.0024)
		tmp = Float64(fma(t_1, Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * fma(-0.0625, sin(x), y)), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.00355], t$95$2, If[LessEqual[y, 0.0024], N[(N[(t$95$1 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0035500000000000002 or 0.00239999999999999979 < y

   1. Initial program 99.1%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 60.6

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

   9. Simplified 60.6%

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

   if -0.0035500000000000002 < y < 0.00239999999999999979

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sin.f64 99.4

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

   9. Simplified 99.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00355:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 0.0024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(t\_0, \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), t\_1 \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma t_0 (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y)))) 2.0)
          (+
           3.0
           (*
            3.0
            (fma
             (cos x)
             (fma (sqrt 5.0) 0.5 -0.5)
             (* t_1 (* (cos y) 0.5))))))))
   (if (<= y -53000000.0)
     t_2
     (if (<= y 0.00025)
       (/
        (fma
         t_0
         (* (sqrt 2.0) (* (fma (sin x) -0.0625 (sin y)) (+ (cos x) -1.0)))
         2.0)
        (* 3.0 (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_1) 1.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(t_0, (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 2.0) / (3.0 + (3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), (t_1 * (cos(y) * 0.5)))));
	double tmp;
	if (y <= -53000000.0) {
		tmp = t_2;
	} else if (y <= 0.00025) {
		tmp = fma(t_0, (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * (cos(x) + -1.0))), 2.0) / (3.0 * fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_1), 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(t_0, Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(3.0 * fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), Float64(t_1 * Float64(cos(y) * 0.5))))))
	tmp = 0.0
	if (y <= -53000000.0)
		tmp = t_2;
	elseif (y <= 0.00025)
		tmp = Float64(fma(t_0, Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 * fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_1), 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$1 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -53000000.0], t$95$2, If[LessEqual[y, 0.00025], N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3e7 or 2.5000000000000001e-4 < y

   1. Initial program 99.1%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 60.9

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

   9. Simplified 60.9%

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

   if -5.3e7 < y < 2.5000000000000001e-4

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 98.8

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. rem-square-sqrt N/A

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

      16. unpow2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. unpow2 N/A

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

      19. rem-square-sqrt 98.7

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

   10. Simplified 98.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{5} + -1\\ t_3 := 1 + \cos x \cdot \frac{t\_2}{2}\\ \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(t\_3 + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_2, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(t\_3 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (+ 1.0 (* (cos x) (/ t_2 2.0)))))
   (if (<= y -53000000.0)
     (/
      (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
      (* 3.0 (+ t_3 (* (cos y) (/ t_0 2.0)))))
     (if (<= y 0.00025)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (sqrt 2.0) (* (fma (sin x) -0.0625 (sin y)) (+ (cos x) -1.0)))
         2.0)
        (* 3.0 (fma 0.5 (fma (cos x) t_2 t_0) 1.0)))
       (/
        (fma (* -0.0625 t_1) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (* 3.0 (+ t_3 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	double tmp;
	if (y <= -53000000.0) {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / (3.0 * (t_3 + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 0.00025) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * (cos(x) + -1.0))), 2.0) / (3.0 * fma(0.5, fma(cos(x), t_2, t_0), 1.0));
	} else {
		tmp = fma((-0.0625 * t_1), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 * (t_3 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0)))
	tmp = 0.0
	if (y <= -53000000.0)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 0.00025)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 * fma(0.5, fma(cos(x), t_2, t_0), 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -53000000.0], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00025], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3e7

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 60.9%

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

   if -5.3e7 < y < 2.5000000000000001e-4

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 98.8

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. lower-+.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. unpow2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. rem-square-sqrt N/A

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

      16. unpow2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. unpow2 N/A

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

      19. rem-square-sqrt 98.7

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

   10. Simplified 98.7%

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

   if 2.5000000000000001e-4 < y

   1. Initial program 99.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 60.3

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

   7. Simplified 60.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x + -1\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{5} + -1\\ t_3 := 1 + \cos x \cdot \frac{t\_2}{2}\\ \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(t\_3 + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00122:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, \cos y \cdot t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(t\_3 + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (+ 1.0 (* (cos x) (/ t_2 2.0)))))
   (if (<= y -53000000.0)
     (/
      (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
      (* 3.0 (+ t_3 (* (cos y) (/ t_0 2.0)))))
     (if (<= y 0.00122)
       (/
        (fma
         0.3333333333333333
         (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
         0.6666666666666666)
        (fma 0.5 (fma t_2 (cos x) (* (cos y) t_0)) 1.0))
       (/
        (fma (* -0.0625 t_1) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (* 3.0 (+ t_3 (* (cos y) (/ 2.0 (+ 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	double tmp;
	if (y <= -53000000.0) {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / (3.0 * (t_3 + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 0.00122) {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), (cos(y) * t_0)), 1.0);
	} else {
		tmp = fma((-0.0625 * t_1), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 * (t_3 + (cos(y) * (2.0 / (3.0 + sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0)))
	tmp = 0.0
	if (y <= -53000000.0)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 0.00122)
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), Float64(cos(y) * t_0)), 1.0));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -53000000.0], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00122], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3e7

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 60.9%

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

   if -5.3e7 < y < 0.00121999999999999995

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]

   if 0.00121999999999999995 < y

   1. Initial program 99.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 60.3

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

   7. Simplified 60.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00122:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{2}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00122:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, \cos y \cdot t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (/
          (fma
           (pow (sin y) 2.0)
           (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625))
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0)))))))
   (if (<= y -53000000.0)
     t_2
     (if (<= y 0.00122)
       (/
        (fma
         0.3333333333333333
         (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
         0.6666666666666666)
        (fma 0.5 (fma t_1 (cos x) (* (cos y) t_0)) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	double tmp;
	if (y <= -53000000.0) {
		tmp = t_2;
	} else if (y <= 0.00122) {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), (cos(y) * t_0)), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 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)))))
	tmp = 0.0
	if (y <= -53000000.0)
		tmp = t_2;
	elseif (y <= 0.00122)
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), Float64(cos(y) * t_0)), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $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]}, If[LessEqual[y, -53000000.0], t$95$2, If[LessEqual[y, 0.00122], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.3e7 or 0.00121999999999999995 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Simplified 60.6%

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

   if -5.3e7 < y < 0.00121999999999999995

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.00122:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot \left(\sqrt{2} \cdot t\_2\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot t\_0\right) \cdot t\_2, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot t\_1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -9e-5)
     (/
      (fma 0.3333333333333333 (* t_0 (* (sqrt 2.0) t_2)) 0.6666666666666666)
      (fma
       (cos y)
       (/ 2.0 (+ 3.0 (sqrt 5.0)))
       (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_1 (sqrt 5.0)) -0.5) 3.0))
       (/
        (fma 0.3333333333333333 (* (* (sqrt 2.0) t_0) t_2) 0.6666666666666666)
        (fma 0.5 (fma (+ (sqrt 5.0) -1.0) (cos x) (* (cos y) t_1)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -9e-5) {
		tmp = fma(0.3333333333333333, (t_0 * (sqrt(2.0) * t_2)), 0.6666666666666666) / fma(cos(y), (2.0 / (3.0 + sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * t_0) * t_2), 0.6666666666666666) / fma(0.5, fma((sqrt(5.0) + -1.0), cos(x), (cos(y) * t_1)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -9e-5)
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * Float64(sqrt(2.0) * t_2)), 0.6666666666666666) / fma(cos(y), Float64(2.0 / Float64(3.0 + sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * t_0) * t_2), 0.6666666666666666) / fma(0.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), Float64(cos(y) * t_1)), 1.0));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -9e-5], N[(N[(0.3333333333333333 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000057e-5

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.4%

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

      1. lift-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{\color{blue}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

      11. lower-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \frac{\cos y \cdot \left(9 - {\left(\sqrt{5}\right)}^{2}\right)}{3 + \sqrt{5}}\right)}} \]

   10. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Simplified 98.3%

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

   if 0.00139999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 58.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, \frac{2}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot t\_0\right), 1\right)}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           0.3333333333333333
           (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
           0.6666666666666666)
          (fma 0.5 (fma (+ (sqrt 5.0) -1.0) (cos x) (* (cos y) t_0)) 1.0))))
   (if (<= x -9e-5)
     t_1
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma((sqrt(5.0) + -1.0), cos(x), (cos(y) * t_0)), 1.0);
	double tmp;
	if (x <= -9e-5) {
		tmp = t_1;
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), Float64(cos(y) * t_0)), 1.0))
	tmp = 0.0
	if (x <= -9e-5)
		tmp = t_1;
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-5], t$95$1, If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000057e-5 or 0.00139999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Simplified 98.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\cos y, t\_1 \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t\_0}{t\_2}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.3333333333333333}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
          (+ 0.5 (* -0.5 (cos (+ x x))))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (cos y)
          (* t_1 0.5)
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))
   (if (<= x -9e-5)
     (/ (* 0.3333333333333333 t_0) t_2)
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_1 (sqrt 5.0)) -0.5) 3.0))
       (* t_0 (/ 0.3333333333333333 t_2))))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(cos(y), (t_1 * 0.5), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0));
	double tmp;
	if (x <= -9e-5) {
		tmp = (0.3333333333333333 * t_0) / t_2;
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = t_0 * (0.3333333333333333 / t_2);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(cos(y), Float64(t_1 * 0.5), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))
	tmp = 0.0
	if (x <= -9e-5)
		tmp = Float64(Float64(0.3333333333333333 * t_0) / t_2);
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(t_0 * Float64(0.3333333333333333 / t_2));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 * 0.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-5], N[(N[(0.3333333333333333 * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.3333333333333333 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000057e-5

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.4%

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

   7. Applied egg-rr 58.5%

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

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Simplified 98.3%

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

   if 0.00139999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Simplified 58.6%

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

   7. Applied egg-rr 58.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, t\_0 \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (fma
           (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
           (+ 0.5 (* -0.5 (cos (+ x x))))
           2.0)
          (/
           0.3333333333333333
           (fma
            (cos y)
            (* t_0 0.5)
            (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))))
   (if (<= x -9e-5)
     t_1
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0) * (0.3333333333333333 / fma(cos(y), (t_0 * 0.5), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)));
	double tmp;
	if (x <= -9e-5) {
		tmp = t_1;
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), Float64(t_0 * 0.5), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))))
	tmp = 0.0
	if (x <= -9e-5)
		tmp = t_1;
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-5], t$95$1, If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000057e-5 or 0.00139999999999999999 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 58.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 0.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0 \cdot \left(\sqrt{2} \cdot t\_1\right), 0.6666666666666666\right)}{\frac{2}{3 + \sqrt{5}} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot t\_0\right) \cdot t\_1, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -9e-5)
     (/
      (fma 0.3333333333333333 (* t_0 (* (sqrt 2.0) t_1)) 0.6666666666666666)
      (+
       (/ 2.0 (+ 3.0 (sqrt 5.0)))
       (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma
         3.0
         (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -0.5)
         3.0))
       (/
        (fma 0.3333333333333333 (* (* (sqrt 2.0) t_0) t_1) 0.6666666666666666)
        (fma 0.5 (- (fma (+ (sqrt 5.0) -1.0) (cos x) 3.0) (sqrt 5.0)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -9e-5) {
		tmp = fma(0.3333333333333333, (t_0 * (sqrt(2.0) * t_1)), 0.6666666666666666) / ((2.0 / (3.0 + sqrt(5.0))) + fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * t_0) * t_1), 0.6666666666666666) / fma(0.5, (fma((sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -9e-5)
		tmp = Float64(fma(0.3333333333333333, Float64(t_0 * Float64(sqrt(2.0) * t_1)), 0.6666666666666666) / Float64(Float64(2.0 / Float64(3.0 + sqrt(5.0))) + fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * t_0) * t_1), 0.6666666666666666) / fma(0.5, Float64(fma(Float64(sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0));
	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] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -9e-5], N[(N[(0.3333333333333333 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000057e-5

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 58.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{\color{blue}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

      11. lower-neg.f64 58.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(-\sqrt{5}\right) \cdot \left(-\sqrt{5}\right)}{3 - \color{blue}{\left(-\sqrt{5}\right)}}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 58.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{9 - \left(-\sqrt{5}\right) \cdot \left(-\sqrt{5}\right)}{3 - \left(-\sqrt{5}\right)}}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \frac{9 - {\left(\sqrt{5}\right)}^{2}}{3 + \sqrt{5}}\right)}} \]

   10. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\frac{2}{3 + \sqrt{5}} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]

   if 0.00139999999999999999 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\frac{2}{3 + \sqrt{5}} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, -0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot t\_2\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -9e-5)
     (/
      (fma
       0.3333333333333333
       (* -0.0625 (* t_2 (* (sqrt 2.0) (+ (cos x) -1.0))))
       0.6666666666666666)
      (fma 0.5 (fma (cos x) t_0 t_1) 1.0))
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma 3.0 (fma 0.5 (fma (cos y) t_1 (sqrt 5.0)) -0.5) 3.0))
       (/
        (fma
         0.3333333333333333
         (* (* (sqrt 2.0) t_2) (fma (cos x) -0.0625 0.0625))
         0.6666666666666666)
        (fma 0.5 (- (fma t_0 (cos x) 3.0) (sqrt 5.0)) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -9e-5) {
		tmp = fma(0.3333333333333333, (-0.0625 * (t_2 * (sqrt(2.0) * (cos(x) + -1.0)))), 0.6666666666666666) / fma(0.5, fma(cos(x), t_0, t_1), 1.0);
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = fma(0.3333333333333333, ((sqrt(2.0) * t_2) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, (fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -9e-5)
		tmp = Float64(fma(0.3333333333333333, Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))), 0.6666666666666666) / fma(0.5, fma(cos(x), t_0, t_1), 1.0));
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0));
	else
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * t_2) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, Float64(fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0));
	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]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -9e-5], N[(N[(0.3333333333333333 * N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000057e-5

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\cos x - 1\right)}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\cos x + \color{blue}{-1}\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\cos x + -1\right)}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-cos.f64 59.4

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\color{blue}{\cos x} + -1\right)\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Simplified 59.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\left(\cos x + -1\right)}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Simplified 57.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]

   if 0.00139999999999999999 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           0.3333333333333333
           (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
           0.6666666666666666)
          (fma 0.5 (- (fma (+ (sqrt 5.0) -1.0) (cos x) 3.0) (sqrt 5.0)) 1.0))))
   (if (<= x -9e-5)
     t_0
     (if (<= x 0.0014)
       (/
        (fma (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) (pow (sin y) 2.0) 2.0)
        (fma
         3.0
         (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -0.5)
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, (fma((sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0);
	double tmp;
	if (x <= -9e-5) {
		tmp = t_0;
	} else if (x <= 0.0014) {
		tmp = fma(((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), pow(sin(y), 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, Float64(fma(Float64(sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0))
	tmp = 0.0
	if (x <= -9e-5)
		tmp = t_0;
	elseif (x <= 0.0014)
		tmp = Float64(fma(Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), (sin(y) ^ 2.0), 2.0) / fma(3.0, fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-5], t$95$0, If[LessEqual[x, 0.0014], N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000057e-5 or 0.00139999999999999999 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 58.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}} \]

   if -9.00000000000000057e-5 < x < 0.00139999999999999999

   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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]

   9. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 59.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (* (sqrt 2.0) (pow (sin x) 2.0)) (fma (cos x) -0.0625 0.0625))
   0.6666666666666666)
  (fma 0.5 (- (fma (+ (sqrt 5.0) -1.0) (cos x) 3.0) (sqrt 5.0)) 1.0)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(x), 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, (fma((sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * fma(cos(x), -0.0625, 0.0625)), 0.6666666666666666) / fma(0.5, Float64(fma(Float64(sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 1.0))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

8. Simplified 59.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 1\right)}} \]
9. Add Preprocessing

Alternative 26: 44.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (*
     (cos y)
     (/ (/ (- 9.0 (* (sqrt 5.0) (sqrt 5.0))) (+ 3.0 (sqrt 5.0))) 2.0))))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (((9.0 - (sqrt(5.0) * sqrt(5.0))) / (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 / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (((9.0d0 - (sqrt(5.0d0) * sqrt(5.0d0))) / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (((9.0 - (Math.sqrt(5.0) * Math.sqrt(5.0))) / (3.0 + Math.sqrt(5.0))) / 2.0))));
}

Python

def code(x, y):
	return 2.0 / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (((9.0 - (math.sqrt(5.0) * math.sqrt(5.0))) / (3.0 + math.sqrt(5.0))) / 2.0))))

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(Float64(9.0 - Float64(sqrt(5.0) * sqrt(5.0))) / Float64(3.0 + sqrt(5.0))) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (((9.0 - (sqrt(5.0) * sqrt(5.0))) / (3.0 + sqrt(5.0))) / 2.0))));
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(N[(9.0 - N[(N[Sqrt[5.0], $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   3. flip-+ N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{3 \cdot 3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{\color{blue}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}}{2} \cdot \cos y\right)} \]

   11. lower-neg.f64 61.5

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(-\sqrt{5}\right) \cdot \left(-\sqrt{5}\right)}{3 - \color{blue}{\left(-\sqrt{5}\right)}}}{2} \cdot \cos y\right)} \]

7. Applied egg-rr 61.5%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{9 - \left(-\sqrt{5}\right) \cdot \left(-\sqrt{5}\right)}{3 - \left(-\sqrt{5}\right)}}}{2} \cdot \cos y\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}{3 - \left(\mathsf{neg}\left(\sqrt{5}\right)\right)}}{2} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Simplified 45.9%

    \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \left(-\sqrt{5}\right) \cdot \left(-\sqrt{5}\right)}{3 - \left(-\sqrt{5}\right)}}{2} \cdot \cos y\right)} \]

11. Final simplification 45.9%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2}\right)} \]
12. Add Preprocessing

Alternative 27: 41.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ 0.6666666666666666 \cdot \frac{1}{1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), -0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.6666666666666666
  (/ 1.0 (+ 1.0 (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (sqrt 5.0)) -0.5)))))

C

double code(double x, double y) {
	return 0.6666666666666666 * (1.0 / (1.0 + fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), -0.5)));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 * Float64(1.0 / Float64(1.0 + fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), -0.5))))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 * N[(1.0 / N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $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

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 1\right)} \]

   12. lower-sqrt.f64 43.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

8. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]
9. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)\right) + 1} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \left(\sqrt{5} + -1\right)\right) + 1} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \left(\sqrt{5} + -1\right)\right) + 1} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\color{blue}{\sqrt{5}} + -1\right)\right) + 1} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + -1\right)}\right) + 1} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right)} + 1} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

   8. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}{\frac{2}{3}}}} \]

   9. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \cdot \frac{2}{3}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \cdot \frac{2}{3}} \]

10. Applied egg-rr 43.2%

    \[\leadsto \color{blue}{\frac{1}{1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), -0.5\right)} \cdot 0.6666666666666666} \]

11. Final simplification 43.2%

    \[\leadsto 0.6666666666666666 \cdot \frac{1}{1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), -0.5\right)} \]
12. Add Preprocessing

Alternative 28: 41.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (+ (sqrt 5.0) -1.0)) 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) + -1.0)), 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) + -1.0)), 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 1\right)} \]

   12. lower-sqrt.f64 43.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

8. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]
9. Add Preprocessing

Alternative 29: 41.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, -1 + \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma 0.5 (+ -1.0 (fma (- 3.0 (sqrt 5.0)) (cos y) (sqrt 5.0))) 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(0.5, (-1.0 + fma((3.0 - sqrt(5.0)), cos(y), sqrt(5.0))), 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(0.5, Float64(-1.0 + fma(Float64(3.0 - sqrt(5.0)), cos(y), sqrt(5.0))), 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(0.5 * N[(-1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 1\right)} \]

   12. lower-sqrt.f64 43.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

8. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]
9. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right), 1\right)} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \left(\sqrt{5} + -1\right), 1\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \left(\sqrt{5} + -1\right), 1\right)} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1}, 1\right)} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1}, 1\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \sqrt{5}\right) + -1, 1\right)} \]

   8. lower-fma.f64 43.2

      \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right)} + -1, 1\right)} \]

10. Applied egg-rr 43.2%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right) + -1}, 1\right)} \]

11. Final simplification 43.2%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, -1 + \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1\right)} \]
12. Add Preprocessing

Alternative 30: 40.0% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 1\right)} \]

   12. lower-sqrt.f64 43.2

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

8. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3}} \]
10. Step-by-step derivation

11. Simplified 41.4%

    \[\leadsto \color{blue}{0.3333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))))))