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

Percentage Accurate: 99.3% → 99.4%

Time: 12.9s

Alternatives: 35

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.3

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lift-sqrt.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

11. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625)))
   (* (- (cos x) (cos y)) (sqrt 2.0))
   2.0)
  (fma
   (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) 1.0)
   3.0
   (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return fma(((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625))), ((cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625))), Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.3

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lift-sqrt.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \end{array} \]

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
   (fma
    0.5
    (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
    1.0))))

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 * fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0));
}

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 * fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)))
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[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.2%

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

   1. lift-fma.f64 N/A

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

9. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.2%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 99.2%

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

Alternative 8: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{if}\;y \leq -0.082:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - t\_0\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* y y) (- (* 0.041666666666666664 (* y y)) 0.5))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           2.0
           (*
            (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
            (- (cos x) (cos y))))
          (+
           (* (fma (cos x) (/ t_1 2.0) 1.0) 3.0)
           (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))
   (if (<= y -0.082)
     t_2
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625)))
          (* (- (cos x) t_0) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) t_0)) 1.0))
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((y * y) * ((0.041666666666666664 * (y * y)) - 0.5));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / ((fma(cos(x), (t_1 / 2.0), 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	double tmp;
	if (y <= -0.082) {
		tmp = t_2;
	} else if (y <= 5.5e-23) {
		tmp = (fma(((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625))), ((cos(x) - t_0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * t_0)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5)))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(cos(x) - cos(y)))) / Float64(Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))
	tmp = 0.0
	if (y <= -0.082)
		tmp = t_2;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625))), Float64(Float64(cos(x) - t_0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * t_0)), 1.0)) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = 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[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.082], t$95$2, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0820000000000000034 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.1

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

   3. Applied rewrites 99.1%

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 63.7

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

   11. Applied rewrites 63.7%

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

   if -0.0820000000000000034 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ t_3 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -0.165:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_3 \cdot 0.0625\right) \cdot \left(t\_3 - \sin x \cdot 0.0625\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (+
           (* (fma (cos x) (/ t_1 2.0) 1.0) 3.0)
           (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))
        (t_3 (* y (- 1.0 (* 0.16666666666666666 (* y y))))))
   (if (<= y -0.165)
     t_2
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* (- (sin x) (* t_3 0.0625)) (- t_3 (* (sin x) 0.0625)))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0))
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / ((fma(cos(x), (t_1 / 2.0), 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	double t_3 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double tmp;
	if (y <= -0.165) {
		tmp = t_2;
	} else if (y <= 5.5e-23) {
		tmp = (fma(((sin(x) - (t_3 * 0.0625)) * (t_3 - (sin(x) * 0.0625))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))
	t_3 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	tmp = 0.0
	if (y <= -0.165)
		tmp = t_2;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(t_3 * 0.0625)) * Float64(t_3 - Float64(sin(x) * 0.0625))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(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[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.165], t$95$2, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$3 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.165000000000000008 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.1

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

   3. Applied rewrites 99.1%

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lift-sin.f64 63.7

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

   11. Applied rewrites 63.7%

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

   if -0.165000000000000008 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       7. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{\mathsf{fma}\left(\left(\sin y \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_3}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.165:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_2 \cdot 0.0625\right) \cdot \left(t\_2 - \sin x \cdot 0.0625\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* y (- 1.0 (* 0.16666666666666666 (* y y)))))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4
         (/
          (fma
           (* (* (sin y) (- (sin x) (* (sin y) 0.0625))) t_0)
           (sqrt 2.0)
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (/ t_3 2.0) (cos x))) (* (/ t_1 2.0) (cos y)))))))
   (if (<= y -0.165)
     t_4
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* (- (sin x) (* t_2 0.0625)) (- t_2 (* (sin x) 0.0625)))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_3 (cos x) (* t_1 (cos y))) 1.0))
        0.3333333333333333)
       t_4))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(((sin(y) * (sin(x) - (sin(y) * 0.0625))) * t_0), sqrt(2.0), 2.0) / (3.0 * ((1.0 + ((t_3 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.165) {
		tmp = t_4;
	} else if (y <= 5.5e-23) {
		tmp = (fma(((sin(x) - (t_2 * 0.0625)) * (t_2 - (sin(x) * 0.0625))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(fma(Float64(Float64(sin(y) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * t_0), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_3 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.165)
		tmp = t_4;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(t_2 * 0.0625)) * Float64(t_2 - Float64(sin(x) * 0.0625))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.165], t$95$4, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$2 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.165000000000000008 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -0.165000000000000008 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       7. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_1 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ t_2 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)\\ t_3 := \frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \sin y, t\_0, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -0.165:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_1 \cdot 0.0625\right) \cdot \left(t\_1 - \sin x \cdot 0.0625\right), t\_0, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) (cos y)) (sqrt 2.0)))
        (t_1 (* y (- 1.0 (* 0.16666666666666666 (* y y)))))
        (t_2
         (fma
          0.5
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.0))
        (t_3
         (*
          (/ (fma (* (- (sin x) (* (sin y) 0.0625)) (sin y)) t_0 2.0) t_2)
          0.3333333333333333)))
   (if (<= y -0.165)
     t_3
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* (- (sin x) (* t_1 0.0625)) (- t_1 (* (sin x) 0.0625)))
          t_0
          2.0)
         t_2)
        0.3333333333333333)
       t_3))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - cos(y)) * sqrt(2.0);
	double t_1 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double t_2 = fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0);
	double t_3 = (fma(((sin(x) - (sin(y) * 0.0625)) * sin(y)), t_0, 2.0) / t_2) * 0.3333333333333333;
	double tmp;
	if (y <= -0.165) {
		tmp = t_3;
	} else if (y <= 5.5e-23) {
		tmp = (fma(((sin(x) - (t_1 * 0.0625)) * (t_1 - (sin(x) * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_1 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	t_2 = fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)
	t_3 = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * sin(y)), t_0, 2.0) / t_2) * 0.3333333333333333)
	tmp = 0.0
	if (y <= -0.165)
		tmp = t_3;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(t_1 * 0.0625)) * Float64(t_1 - Float64(sin(x) * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[y, -0.165], t$95$3, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$1 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.165000000000000008 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 24.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.5

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

   10. Applied rewrites 63.5%

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

   if -0.165000000000000008 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       7. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - \cos y\right) \cdot \sqrt{2}\\ t_1 := x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ t_2 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)\\ t_3 := \frac{\mathsf{fma}\left(\sin x \cdot \left(\sin y - \sin x \cdot 0.0625\right), t\_0, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -0.105:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(t\_1 - \sin y \cdot 0.0625\right) \cdot \left(\sin y - t\_1 \cdot 0.0625\right), t\_0, 2\right)}{t\_2} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) (cos y)) (sqrt 2.0)))
        (t_1 (* x (- 1.0 (* 0.16666666666666666 (* x x)))))
        (t_2
         (fma
          0.5
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.0))
        (t_3
         (*
          (/ (fma (* (sin x) (- (sin y) (* (sin x) 0.0625))) t_0 2.0) t_2)
          0.3333333333333333)))
   (if (<= x -0.105)
     t_3
     (if (<= x 0.072)
       (*
        (/
         (fma
          (* (- t_1 (* (sin y) 0.0625)) (- (sin y) (* t_1 0.0625)))
          t_0
          2.0)
         t_2)
        0.3333333333333333)
       t_3))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - cos(y)) * sqrt(2.0);
	double t_1 = x * (1.0 - (0.16666666666666666 * (x * x)));
	double t_2 = fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0);
	double t_3 = (fma((sin(x) * (sin(y) - (sin(x) * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333;
	double tmp;
	if (x <= -0.105) {
		tmp = t_3;
	} else if (x <= 0.072) {
		tmp = (fma(((t_1 - (sin(y) * 0.0625)) * (sin(y) - (t_1 * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))
	t_1 = Float64(x * Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))))
	t_2 = fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)
	t_3 = Float64(Float64(fma(Float64(sin(x) * Float64(sin(y) - Float64(sin(x) * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -0.105)
		tmp = t_3;
	elseif (x <= 0.072)
		tmp = Float64(Float64(fma(Float64(Float64(t_1 - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(t_1 * 0.0625))), t_0, 2.0) / t_2) * 0.3333333333333333);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -0.105], t$95$3, If[LessEqual[x, 0.072], N[(N[(N[(N[(N[(t$95$1 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$1 * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.104999999999999996 or 0.0719999999999999946 < 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -0.104999999999999996 < x < 0.0719999999999999946

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 61.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-*.f64 99.4

         \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       6. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right) - \sin y \cdot \frac{1}{16}\right) \cdot \left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       7. lift-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\sin y \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ t_3 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -0.17:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sin x - t\_3 \cdot 0.0625\right) \cdot \left(t\_3 - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (* (sin y) (- (sin x) (* (sin y) 0.0625))) (- 1.0 (cos y)))
           (sqrt 2.0)
           2.0)
          (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y))))))
        (t_3 (* y (- 1.0 (* 0.16666666666666666 (* y y))))))
   (if (<= y -0.17)
     t_2
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* (- (sin x) (* t_3 0.0625)) (- t_3 (* (sin x) 0.0625)))
          (* (- (cos x) (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_1 (cos x) (* t_0 (cos y))) 1.0))
        0.3333333333333333)
       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(((sin(y) * (sin(x) - (sin(y) * 0.0625))) * (1.0 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double t_3 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double tmp;
	if (y <= -0.17) {
		tmp = t_2;
	} else if (y <= 5.5e-23) {
		tmp = (fma(((sin(x) - (t_3 * 0.0625)) * (t_3 - (sin(x) * 0.0625))), ((cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), (t_0 * cos(y))), 1.0)) * 0.3333333333333333;
	} 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(Float64(Float64(sin(y) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	t_3 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	tmp = 0.0
	if (y <= -0.17)
		tmp = t_2;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(Float64(sin(x) - Float64(t_3 * 0.0625)) * Float64(t_3 - Float64(sin(x) * 0.0625))), Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), Float64(t_0 * cos(y))), 1.0)) * 0.3333333333333333);
	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[(N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.17], t$95$2, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(t$95$3 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.170000000000000012 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -0.170000000000000012 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin y - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-*.f64 99.5

         \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   11. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot {y}^{2}\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(y \cdot \left(1 - \frac{1}{6} \cdot \left(y \cdot y\right)\right) - \sin x \cdot \frac{1}{16}\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

       7. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\sin x - \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot 0.0625\right) \cdot \left(y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right) - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 - 0.5 \cdot \left(y \cdot y\right)\\ t_2 := \sin x - \sin y \cdot 0.0625\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{\mathsf{fma}\left(\left(\sin y \cdot t\_2\right) \cdot \left(1 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_3}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.015:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - t\_1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_0 \cdot t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- 1.0 (* 0.5 (* y y))))
        (t_2 (- (sin x) (* (sin y) 0.0625)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4
         (/
          (fma (* (* (sin y) t_2) (- 1.0 (cos y))) (sqrt 2.0) 2.0)
          (*
           3.0
           (+ (+ 1.0 (* (/ t_3 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -0.015)
     t_4
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* t_2 (- (sin y) (* (sin x) 0.0625)))
          (* (- (cos x) t_1) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_3 (cos x) (* t_0 t_1)) 1.0))
        0.3333333333333333)
       t_4))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 - (0.5 * (y * y));
	double t_2 = sin(x) - (sin(y) * 0.0625);
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = fma(((sin(y) * t_2) * (1.0 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + ((t_3 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.015) {
		tmp = t_4;
	} else if (y <= 5.5e-23) {
		tmp = (fma((t_2 * (sin(y) - (sin(x) * 0.0625))), ((cos(x) - t_1) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), (t_0 * t_1)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(y * y)))
	t_2 = Float64(sin(x) - Float64(sin(y) * 0.0625))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(fma(Float64(Float64(sin(y) * t_2) * Float64(1.0 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_3 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.015)
		tmp = t_4;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(t_2 * Float64(sin(y) - Float64(sin(x) * 0.0625))), Float64(Float64(cos(x) - t_1) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), Float64(t_0 * t_1)), 1.0)) * 0.3333333333333333);
	else
		tmp = t_4;
	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[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.015], t$95$4, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.014999999999999999 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lift-cos.f64 60.7

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

   7. Applied rewrites 60.7%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -0.014999999999999999 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in y around 0

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

       1. fp-cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 99.5

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

   13. Applied rewrites 99.5%

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

Alternative 15: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos \left(x + x\right)\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_0}{2}\\ t_5 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{2 + \left(\left(\left(0.5 - t\_1 \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_2}{\mathsf{fma}\left(\cos x, t\_4, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(1.00390625, \sin y, x \cdot \left(x \cdot \left(\sin y \cdot -0.16731770833333334\right) - 0.0625\right)\right)\right) \cdot \left(t\_5 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_4 \cdot t\_5\right) + \frac{t\_3}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot t\_1\right), t\_2 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_3 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (cos (+ x x)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_0 2.0))
        (t_5 (fma (* x x) -0.5 1.0)))
   (if (<= x -0.034)
     (/
      (+ 2.0 (* (* (* (- 0.5 (* t_1 0.5)) -0.0625) (sqrt 2.0)) t_2))
      (+ (* (fma (cos x) t_4 1.0) 3.0) (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
     (if (<= x 0.0165)
       (/
        (fma
         (*
          (fma
           -0.0625
           (- 0.5 (* 0.5 (cos (+ y y))))
           (*
            x
            (fma
             1.00390625
             (sin y)
             (* x (- (* x (* (sin y) -0.16731770833333334)) 0.0625)))))
          (- t_5 (cos y)))
         (sqrt 2.0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_4 t_5)) (* (/ t_3 2.0) (cos y)))))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 t_1))) (* t_2 (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_0 (cos x) (* t_3 (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos((x + x));
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_0 / 2.0;
	double t_5 = fma((x * x), -0.5, 1.0);
	double tmp;
	if (x <= -0.034) {
		tmp = (2.0 + ((((0.5 - (t_1 * 0.5)) * -0.0625) * sqrt(2.0)) * t_2)) / ((fma(cos(x), t_4, 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	} else if (x <= 0.0165) {
		tmp = fma((fma(-0.0625, (0.5 - (0.5 * cos((y + y)))), (x * fma(1.00390625, sin(y), (x * ((x * (sin(y) * -0.16731770833333334)) - 0.0625))))) * (t_5 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + (t_4 * t_5)) + ((t_3 / 2.0) * cos(y))));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * t_1))), (t_2 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), (t_3 * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = cos(Float64(x + x))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_0 / 2.0)
	t_5 = fma(Float64(x * x), -0.5, 1.0)
	tmp = 0.0
	if (x <= -0.034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(Float64(0.5 - Float64(t_1 * 0.5)) * -0.0625) * sqrt(2.0)) * t_2)) / Float64(Float64(fma(cos(x), t_4, 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	elseif (x <= 0.0165)
		tmp = Float64(fma(Float64(fma(-0.0625, Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(x * fma(1.00390625, sin(y), Float64(x * Float64(Float64(x * Float64(sin(y) * -0.16731770833333334)) - 0.0625))))) * Float64(t_5 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_4 * t_5)) + Float64(Float64(t_3 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * t_1))), Float64(t_2 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(t_3 * cos(y))), 1.0)) * 0.3333333333333333);
	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[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.034], N[(N[(2.0 + N[(N[(N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0165], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.00390625 * N[Sin[y], $MachinePrecision] + N[(x * N[(N[(x * N[(N[Sin[y], $MachinePrecision] * -0.16731770833333334), $MachinePrecision]), $MachinePrecision] - 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$5 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$4 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.034000000000000002

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. count-2-rev N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-+.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. lift-sqrt.f64 60.4

           \[\leadsto \frac{2 + \left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 60.4%

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

   if -0.034000000000000002 < x < 0.016500000000000001

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \left(\frac{1}{256} \cdot \sin y + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right) - \frac{1}{16}\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right) + x \cdot \left(\sin y + \left(\frac{1}{256} \cdot \sin y + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right) - \frac{1}{16}\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. sqr-sin-a-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) + x \cdot \left(\sin y + \left(\frac{1}{256} \cdot \sin y + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right) - \frac{1}{16}\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right), x \cdot \left(\sin y + \left(\frac{1}{256} \cdot \sin y + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \sin y + \frac{-1}{1536} \cdot \sin y\right) - \frac{1}{16}\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(1.00390625, \sin y, x \cdot \left(x \cdot \left(\sin y \cdot -0.16731770833333334\right) - 0.0625\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 16: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_2}{2}\\ t_4 := x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ t_5 := \cos \left(x + x\right)\\ t_6 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{2 + \left(\left(\left(0.5 - t\_5 \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0}{\mathsf{fma}\left(\cos x, t\_3, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.0165:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - t\_4 \cdot 0.0625\right) \cdot \left(t\_4 - \sin y \cdot 0.0625\right)\right) \cdot \left(t\_1 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_3 \cdot t\_1\right) + \frac{t\_6}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot t\_5\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_6 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (* x x) -0.5 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_2 2.0))
        (t_4 (* x (- 1.0 (* 0.16666666666666666 (* x x)))))
        (t_5 (cos (+ x x)))
        (t_6 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.034)
     (/
      (+ 2.0 (* (* (* (- 0.5 (* t_5 0.5)) -0.0625) (sqrt 2.0)) t_0))
      (+ (* (fma (cos x) t_3 1.0) 3.0) (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
     (if (<= x 0.0165)
       (/
        (fma
         (*
          (* (- (sin y) (* t_4 0.0625)) (- t_4 (* (sin y) 0.0625)))
          (- t_1 (cos y)))
         (sqrt 2.0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_3 t_1)) (* (/ t_6 2.0) (cos y)))))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 t_5))) (* t_0 (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_2 (cos x) (* t_6 (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((x * x), -0.5, 1.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_2 / 2.0;
	double t_4 = x * (1.0 - (0.16666666666666666 * (x * x)));
	double t_5 = cos((x + x));
	double t_6 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.034) {
		tmp = (2.0 + ((((0.5 - (t_5 * 0.5)) * -0.0625) * sqrt(2.0)) * t_0)) / ((fma(cos(x), t_3, 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	} else if (x <= 0.0165) {
		tmp = fma((((sin(y) - (t_4 * 0.0625)) * (t_4 - (sin(y) * 0.0625))) * (t_1 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + (t_3 * t_1)) + ((t_6 / 2.0) * cos(y))));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * t_5))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(x), (t_6 * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(x * x), -0.5, 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 / 2.0)
	t_4 = Float64(x * Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))))
	t_5 = cos(Float64(x + x))
	t_6 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(Float64(0.5 - Float64(t_5 * 0.5)) * -0.0625) * sqrt(2.0)) * t_0)) / Float64(Float64(fma(cos(x), t_3, 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	elseif (x <= 0.0165)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(t_4 * 0.0625)) * Float64(t_4 - Float64(sin(y) * 0.0625))) * Float64(t_1 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * t_1)) + Float64(Float64(t_6 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * t_5))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(x), Float64(t_6 * cos(y))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], N[(N[(2.0 + N[(N[(N[(N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0165], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(t$95$4 * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$6 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$6 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.034000000000000002

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. count-2-rev N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-+.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. lift-sqrt.f64 60.4

           \[\leadsto \frac{2 + \left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 60.4%

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

   if -0.034000000000000002 < x < 0.016500000000000001

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(x \cdot \left(1 - \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - \frac{1}{6} \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{16}\right) \cdot \left(x \cdot \left(1 - \frac{1}{6} \cdot {x}^{2}\right) - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right) \cdot \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.0625\right) \cdot \left(x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right) - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 17: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_1 := \cos x - \cos y\\ t_2 := \cos \left(x + x\right)\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{t\_3}{2}\\ t_5 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{2 + \left(\left(\left(0.5 - t\_2 \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_1}{\mathsf{fma}\left(\cos x, t\_4, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - x \cdot 0.0625\right) \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t\_0 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_4 \cdot t\_0\right) + \frac{t\_5}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot t\_2\right), t\_1 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_5 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.5 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (cos (+ x x)))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (/ t_3 2.0))
        (t_5 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.005)
     (/
      (+ 2.0 (* (* (* (- 0.5 (* t_2 0.5)) -0.0625) (sqrt 2.0)) t_1))
      (+ (* (fma (cos x) t_4 1.0) 3.0) (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
     (if (<= x 0.0135)
       (/
        (fma
         (*
          (* (- (sin y) (* x 0.0625)) (- x (* (sin y) 0.0625)))
          (- t_0 (cos y)))
         (sqrt 2.0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_4 t_0)) (* (/ t_5 2.0) (cos y)))))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 t_2))) (* t_1 (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_3 (cos x) (* t_5 (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.5, 1.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = cos((x + x));
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = t_3 / 2.0;
	double t_5 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.005) {
		tmp = (2.0 + ((((0.5 - (t_2 * 0.5)) * -0.0625) * sqrt(2.0)) * t_1)) / ((fma(cos(x), t_4, 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	} else if (x <= 0.0135) {
		tmp = fma((((sin(y) - (x * 0.0625)) * (x - (sin(y) * 0.0625))) * (t_0 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + (t_4 * t_0)) + ((t_5 / 2.0) * cos(y))));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * t_2))), (t_1 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), (t_5 * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), -0.5, 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = cos(Float64(x + x))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(t_3 / 2.0)
	t_5 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(Float64(0.5 - Float64(t_2 * 0.5)) * -0.0625) * sqrt(2.0)) * t_1)) / Float64(Float64(fma(cos(x), t_4, 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(x * 0.0625)) * Float64(x - Float64(sin(y) * 0.0625))) * Float64(t_0 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_4 * t_0)) + Float64(Float64(t_5 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * t_2))), Float64(t_1 * sqrt(2.0)), 2.0) / fma(0.5, fma(t_3, cos(x), Float64(t_5 * cos(y))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(N[(2.0 + N[(N[(N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       8. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       11. count-2-rev N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       12. lower-+.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \frac{-1}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       13. lift-sqrt.f64 60.4

           \[\leadsto \frac{2 + \left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   11. Applied rewrites 60.4%

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

   if -0.0050000000000000001 < x < 0.0134999999999999998

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

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

   14. Taylor expanded in x around 0

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

   16. Applied rewrites 99.5%

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

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 18: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{t\_1}{2}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ t_4 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, t\_2, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - x \cdot 0.0625\right) \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t\_0 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_0\right) + \frac{t\_4}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_4 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.5 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (/ t_1 2.0))
        (t_3 (- 0.5 (* 0.5 (cos (+ x x)))))
        (t_4 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.005)
     (/
      (- 2.0 (* 0.0625 (* t_3 (* (sqrt 2.0) (- (cos x) 1.0)))))
      (+
       (* (fma (cos x) t_2 1.0) 3.0)
       (* (* (cos y) (/ 4.0 (* (+ (sqrt 5.0) 3.0) 2.0))) 3.0)))
     (if (<= x 0.0135)
       (/
        (fma
         (*
          (* (- (sin y) (* x 0.0625)) (- x (* (sin y) 0.0625)))
          (- t_0 (cos y)))
         (sqrt 2.0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_2 t_0)) (* (/ t_4 2.0) (cos y)))))
       (*
        (/
         (fma (* -0.0625 t_3) (* (- (cos x) (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma t_1 (cos x) (* t_4 (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.5, 1.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = t_1 / 2.0;
	double t_3 = 0.5 - (0.5 * cos((x + x)));
	double t_4 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.005) {
		tmp = (2.0 - (0.0625 * (t_3 * (sqrt(2.0) * (cos(x) - 1.0))))) / ((fma(cos(x), t_2, 1.0) * 3.0) + ((cos(y) * (4.0 / ((sqrt(5.0) + 3.0) * 2.0))) * 3.0));
	} else if (x <= 0.0135) {
		tmp = fma((((sin(y) - (x * 0.0625)) * (x - (sin(y) * 0.0625))) * (t_0 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + (t_2 * t_0)) + ((t_4 / 2.0) * cos(y))));
	} else {
		tmp = (fma((-0.0625 * t_3), ((cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), (t_4 * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), -0.5, 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(t_1 / 2.0)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))
	t_4 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(Float64(fma(cos(x), t_2, 1.0) * 3.0) + Float64(Float64(cos(y) * Float64(4.0 / Float64(Float64(sqrt(5.0) + 3.0) * 2.0))) * 3.0)));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(x * 0.0625)) * Float64(x - Float64(sin(y) * 0.0625))) * Float64(t_0 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_0)) + Float64(Float64(t_4 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(x), Float64(t_4 * cos(y))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower--.f64 N/A

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

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3} \]

   if -0.0050000000000000001 < x < 0.0134999999999999998

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

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

   14. Taylor expanded in x around 0

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

   16. Applied rewrites 99.5%

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

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 60.3

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

   10. Applied rewrites 60.3%

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

Alternative 19: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_1 := 2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_2}{2}\\ t_4 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.005:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\cos x, t\_3, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(\sin y - x \cdot 0.0625\right) \cdot \left(x - \sin y \cdot 0.0625\right)\right) \cdot \left(t\_0 - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_3 \cdot t\_0\right) + \frac{t\_4}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_4 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (* x x) -0.5 1.0))
        (t_1
         (-
          2.0
          (*
           0.0625
           (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_2 2.0))
        (t_4 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.005)
     (/
      t_1
      (+
       (* (fma (cos x) t_3 1.0) 3.0)
       (* (* (cos y) (/ 4.0 (* (+ (sqrt 5.0) 3.0) 2.0))) 3.0)))
     (if (<= x 0.0135)
       (/
        (fma
         (*
          (* (- (sin y) (* x 0.0625)) (- x (* (sin y) 0.0625)))
          (- t_0 (cos y)))
         (sqrt 2.0)
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_3 t_0)) (* (/ t_4 2.0) (cos y)))))
       (*
        (/ t_1 (fma 0.5 (fma t_2 (cos x) (* t_4 (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((x * x), -0.5, 1.0);
	double t_1 = 2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_2 / 2.0;
	double t_4 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.005) {
		tmp = t_1 / ((fma(cos(x), t_3, 1.0) * 3.0) + ((cos(y) * (4.0 / ((sqrt(5.0) + 3.0) * 2.0))) * 3.0));
	} else if (x <= 0.0135) {
		tmp = fma((((sin(y) - (x * 0.0625)) * (x - (sin(y) * 0.0625))) * (t_0 - cos(y))), sqrt(2.0), 2.0) / (3.0 * ((1.0 + (t_3 * t_0)) + ((t_4 / 2.0) * cos(y))));
	} else {
		tmp = (t_1 / fma(0.5, fma(t_2, cos(x), (t_4 * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(x * x), -0.5, 1.0)
	t_1 = Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 / 2.0)
	t_4 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.005)
		tmp = Float64(t_1 / Float64(Float64(fma(cos(x), t_3, 1.0) * 3.0) + Float64(Float64(cos(y) * Float64(4.0 / Float64(Float64(sqrt(5.0) + 3.0) * 2.0))) * 3.0)));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(Float64(sin(y) - Float64(x * 0.0625)) * Float64(x - Float64(sin(y) * 0.0625))) * Float64(t_0 - cos(y))), sqrt(2.0), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * t_0)) + Float64(Float64(t_4 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(t_1 / fma(0.5, fma(t_2, cos(x), Float64(t_4 * cos(y))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.005], N[(t$95$1 / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0050000000000000001

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower--.f64 N/A

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

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3} \]

   if -0.0050000000000000001 < x < 0.0134999999999999998

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

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

   14. Taylor expanded in x around 0

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

   16. Applied rewrites 99.5%

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

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower--.f64 N/A

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

   10. Applied rewrites 60.3%

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

Alternative 20: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3\\ t_1 := \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_0 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}\\ \mathbf{if}\;y \leq -420000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0))
        (t_1
         (/
          (fma
           (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (+ t_0 (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))
   (if (<= y -420000000.0)
     t_1
     (if (<= y 5.5e-23)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ t_0 (* (* (cos y) (/ 4.0 (* (+ (sqrt 5.0) 3.0) 2.0))) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0;
	double t_1 = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (t_0 + (6.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	double tmp;
	if (y <= -420000000.0) {
		tmp = t_1;
	} else if (y <= 5.5e-23) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (t_0 + ((cos(y) * (4.0 / ((sqrt(5.0) + 3.0) * 2.0))) * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0)
	t_1 = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(t_0 + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))
	tmp = 0.0
	if (y <= -420000000.0)
		tmp = t_1;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(t_0 + Float64(Float64(cos(y) * Float64(4.0 / Float64(Float64(sqrt(5.0) + 3.0) * 2.0))) * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -420000000.0], t$95$1, If[LessEqual[y, 5.5e-23], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e8 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.1

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

   3. Applied rewrites 99.1%

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 60.3%

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

   if -4.2e8 < y < 5.5000000000000001e-23

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower--.f64 N/A

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

   8. Applied rewrites 97.7%

      \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\\ t_1 := \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -420000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0)
          (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))
        (t_1
         (/
          (fma
           (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          t_0)))
   (if (<= y -420000000.0)
     t_1
     (if (<= y 5.5e-23)
       (/
        (fma
         (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
         (* (- (cos x) 1.0) (sqrt 2.0))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = (fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))));
	double t_1 = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0;
	double tmp;
	if (y <= -420000000.0) {
		tmp = t_1;
	} else if (y <= 5.5e-23) {
		tmp = fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))
	t_1 = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0)
	tmp = 0.0
	if (y <= -420000000.0)
		tmp = t_1;
	elseif (y <= 5.5e-23)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -420000000.0], t$95$1, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e8 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.1

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

   3. Applied rewrites 99.1%

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. lower-fma.f64 N/A

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

   11. Applied rewrites 60.3%

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

   if -4.2e8 < y < 5.5000000000000001e-23

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 99.5

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 97.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := 3 + \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot -0.0625, t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{t\_1}}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_2, 1\right), 3, \frac{6 \cdot \cos y}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_3\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (cos (+ x x))))
   (if (<= x -3.4e-6)
     (/
      (fma (* (- 0.5 (* t_3 0.5)) -0.0625) (* t_0 (sqrt 2.0)) 2.0)
      (+ (* (fma (cos x) (/ t_2 2.0) 1.0) 3.0) (* 6.0 (/ (cos y) t_1))))
     (if (<= x 0.0135)
       (/
        (fma
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (fma 0.5 t_2 1.0) 3.0 (/ (* 6.0 (cos y)) t_1)))
       (*
        (/
         (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_3)) (* (sqrt 2.0) t_0))))
         (fma 0.5 (fma t_2 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 3.0 + sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = cos((x + x));
	double tmp;
	if (x <= -3.4e-6) {
		tmp = fma(((0.5 - (t_3 * 0.5)) * -0.0625), (t_0 * sqrt(2.0)), 2.0) / ((fma(cos(x), (t_2 / 2.0), 1.0) * 3.0) + (6.0 * (cos(y) / t_1)));
	} else if (x <= 0.0135) {
		tmp = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, ((6.0 * cos(y)) / t_1));
	} else {
		tmp = ((2.0 - (0.0625 * ((0.5 - (0.5 * t_3)) * (sqrt(2.0) * t_0)))) / fma(0.5, fma(t_2, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(3.0 + sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * -0.0625), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(Float64(fma(cos(x), Float64(t_2 / 2.0), 1.0) * 3.0) + Float64(6.0 * Float64(cos(y) / t_1))));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / t_1)));
	else
		tmp = Float64(Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_3)) * Float64(sqrt(2.0) * t_0)))) / fma(0.5, fma(t_2, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e-6], N[(N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000006e-6

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-sqrt.f64 99.0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 60.4%

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

   if -3.40000000000000006e-6 < x < 0.0134999999999999998

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. count-2-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      15. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 60.3%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (-
            2.0
            (*
             0.0625
             (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
           (fma 0.5 (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0))
          0.3333333333333333)))
   (if (<= x -3.4e-6)
     t_1
     (if (<= x 0.0135)
       (/
        (fma
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (fma 0.5 t_0 1.0) 3.0 (/ (* 6.0 (cos y)) (+ 3.0 (sqrt 5.0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(0.5, fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -3.4e-6) {
		tmp = t_1;
	} else if (x <= 0.0135) {
		tmp = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, ((6.0 * cos(y)) / (3.0 + sqrt(5.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(0.5, fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = t_1;
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], t$95$1, If[LessEqual[x, 0.0135], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000006e-6 or 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   7. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right), \left(\cos x - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. unpow2 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. count-2-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      15. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 60.4%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   if -3.40000000000000006e-6 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (fma
           (* -0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (- 1.0 (cos y))))
           (sqrt 2.0)
           2.0)
          (*
           3.0
           (fma 0.5 (fma t_0 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0)))))
   (if (<= y -1.15e-6)
     t_1
     (if (<= y 5.5e-23)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0)
         (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((-0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (1.0 - cos(y)))), sqrt(2.0), 2.0) / (3.0 * fma(0.5, fma(t_0, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0));
	double tmp;
	if (y <= -1.15e-6) {
		tmp = t_1;
	} else if (y <= 5.5e-23) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(fma(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(1.0 - cos(y)))), sqrt(2.0), 2.0) / Float64(3.0 * fma(0.5, fma(t_0, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)))
	tmp = 0.0
	if (y <= -1.15e-6)
		tmp = t_1;
	elseif (y <= 5.5e-23)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e-6], t$95$1, If[LessEqual[y, 5.5e-23], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e-6 or 5.5000000000000001e-23 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) \cdot \sqrt{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right), \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 50.6

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 50.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 51.4%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. sqr-sin-a-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. count-2-rev N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. lower-+.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. lift--.f64 49.7

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 49.7%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. Taylor expanded in x around inf

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   15. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \color{blue}{1}\right)} \]

       2. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)}, 1\right)} \]

   16. Applied rewrites 60.2%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}} \]

   if -1.15e-6 < y < 5.5000000000000001e-23

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{\mathsf{fma}\left(3, 1 - -0.5 \cdot \left(\cos x \cdot t\_2\right), 6 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_2, 1\right), 3, \frac{6 \cdot \cos y}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_2 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -6.2e-5)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_1))))
      (fma 3.0 (- 1.0 (* -0.5 (* (cos x) t_2))) (* 6.0 (/ 1.0 t_0))))
     (if (<= x 0.0135)
       (/
        (fma
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (fma 0.5 t_2 1.0) 3.0 (/ (* 6.0 (cos y)) t_0)))
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_1 (sqrt 2.0))
          2.0)
         (fma 0.5 (+ 3.0 (- (* t_2 (cos x)) (sqrt 5.0))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -6.2e-5) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_1)))) / fma(3.0, (1.0 - (-0.5 * (cos(x) * t_2))), (6.0 * (1.0 / t_0)));
	} else if (x <= 0.0135) {
		tmp = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, ((6.0 * cos(y)) / t_0));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_1 * sqrt(2.0)), 2.0) / fma(0.5, (3.0 + ((t_2 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -6.2e-5)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_1)))) / fma(3.0, Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_2))), Float64(6.0 * Float64(1.0 / t_0))));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_2, 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_1 * sqrt(2.0)), 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_2 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	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[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e-5], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000027e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}} \]

   8. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, 1 - -0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right), 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   if -6.20000000000000027e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 + \sqrt{5}\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \frac{6}{t\_1}\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 1\right), 3, \frac{6 \cdot \cos y}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_2, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -6.2e-5)
     (/
      (fma (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) t_2 2.0)
      (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (/ 6.0 t_1)))
     (if (<= x 0.0135)
       (/
        (fma
         (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (fma 0.5 t_0 1.0) 3.0 (/ (* 6.0 (cos y)) t_1)))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_2 2.0)
         (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
        0.3333333333333333)))))

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 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -6.2e-5) {
		tmp = fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), t_2, 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, (6.0 / t_1));
	} else if (x <= 0.0135) {
		tmp = fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, ((6.0 * cos(y)) / t_1));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_2, 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	}
	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 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -6.2e-5)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), t_2, 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(6.0 / t_1)));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(0.5, t_0, 1.0), 3.0, Float64(Float64(6.0 * cos(y)) / t_1)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_2, 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	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[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-5], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(6.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(6.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000027e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 59.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \frac{6}{3 + \sqrt{5}}\right)}} \]

   if -6.20000000000000027e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right), 3, \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \frac{6}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* (- (cos x) 1.0) (sqrt 2.0))))
   (if (<= x -6.2e-5)
     (/
      (fma (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) t_1 2.0)
      (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (/ 6.0 (+ 3.0 (sqrt 5.0)))))
     (if (<= x 0.0135)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ 1.0 (* 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0)))))
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_1 2.0)
         (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double tmp;
	if (x <= -6.2e-5) {
		tmp = fma(((0.5 - (cos((x + x)) * 0.5)) * -0.0625), t_1, 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, (6.0 / (3.0 + sqrt(5.0))));
	} else if (x <= 0.0135) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (1.0 + (0.5 * fma(cos(y), (3.0 - sqrt(5.0)), t_0))));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_1, 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	tmp = 0.0
	if (x <= -6.2e-5)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), t_1, 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(6.0 / Float64(3.0 + sqrt(5.0)))));
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(1.0 + Float64(0.5 * fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_1, 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-5], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000027e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      6. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{\sqrt{5 \cdot 5}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \sqrt{\color{blue}{25}}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{2 \cdot 2}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{2 \cdot 2}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{4}{\left(\sqrt{5} + 3\right) \cdot 2}\right) \cdot 3}} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \color{blue}{\frac{\cos y}{3 + \sqrt{5}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3 + \sqrt{5}}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{\color{blue}{3} + \sqrt{5}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \color{blue}{\sqrt{5}}}} \]

      5. lift-sqrt.f64 99.0

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}}} \]

   11. Applied rewrites 59.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \frac{6}{3 + \sqrt{5}}\right)}} \]

   if -6.20000000000000027e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right)\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, \color{blue}{3} - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right)\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      9. lift--.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)}} \]

   if 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
          0.3333333333333333)))
   (if (<= x -6e-5)
     t_1
     (if (<= x 0.0135)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ 1.0 (* 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6e-5) {
		tmp = t_1;
	} else if (x <= 0.0135) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (1.0 + (0.5 * fma(cos(y), (3.0 - sqrt(5.0)), t_0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6e-5)
		tmp = t_1;
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(1.0 + Float64(0.5 * fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6e-5], t$95$1, If[LessEqual[x, 0.0135], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000015e-5 or 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -6.00000000000000015e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right)\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, \color{blue}{3} - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right)\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

      9. lift--.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
          0.3333333333333333)))
   (if (<= x -6e-5)
     t_1
     (if (<= x 0.0135)
       (/
        (fma
         (* -0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (- 1.0 (cos y))))
         (sqrt 2.0)
         2.0)
        (* 3.0 (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) t_0) 1.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6e-5) {
		tmp = t_1;
	} else if (x <= 0.0135) {
		tmp = fma((-0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (1.0 - cos(y)))), sqrt(2.0), 2.0) / (3.0 * fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), t_0), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6e-5)
		tmp = t_1;
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(1.0 - cos(y)))), sqrt(2.0), 2.0) / Float64(3.0 * fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), t_0), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6e-5], t$95$1, If[LessEqual[x, 0.0135], N[(N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000015e-5 or 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\cos x \cdot \left(\sqrt{5} - 1\right) + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      16. lift-cos.f64 59.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -6.00000000000000015e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) \cdot \sqrt{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right), \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. sqr-sin-a-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. count-2-rev N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. lower-+.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. lift--.f64 99.0

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   15. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right)} \]

       2. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)}, 1\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \left(3 - \sqrt{5}\right) \cdot \cos y + \left(\color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       7. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos \color{blue}{y}, \sqrt{5} - 1\right), 1\right)} \]

       8. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       9. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       10. lift--.f64 98.8

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   16. Applied rewrites 98.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            -0.0625
            (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* (cos (+ x x)) 0.5)))
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -6e-5)
     t_2
     (if (<= x 0.0135)
       (/
        (fma
         (* -0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (- 1.0 (cos y))))
         (sqrt 2.0)
         2.0)
        (* 3.0 (fma 0.5 (fma t_1 (cos y) t_0) 1.0)))
       t_2))))

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 = (fma(-0.0625, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6e-5) {
		tmp = t_2;
	} else if (x <= 0.0135) {
		tmp = fma((-0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (1.0 - cos(y)))), sqrt(2.0), 2.0) / (3.0 * fma(0.5, fma(t_1, cos(y), t_0), 1.0));
	} else {
		tmp = t_2;
	}
	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 = Float64(Float64(fma(-0.0625, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6e-5)
		tmp = t_2;
	elseif (x <= 0.0135)
		tmp = Float64(fma(Float64(-0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(1.0 - cos(y)))), sqrt(2.0), 2.0) / Float64(3.0 * fma(0.5, fma(t_1, cos(y), t_0), 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0625 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6e-5], t$95$2, If[LessEqual[x, 0.0135], N[(N[(N[(-0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000015e-5 or 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -6.00000000000000015e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right) \cdot \sqrt{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right), \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\cos x - \cos y\right), \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\sin x - \sin y \cdot \frac{1}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. sqr-sin-a-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. count-2-rev N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. lower-+.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lift-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. lift--.f64 99.0

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{\color{blue}{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   15. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right)} \]

       2. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)}, 1\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \left(3 - \sqrt{5}\right) \cdot \cos y + \left(\color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       7. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos \color{blue}{y}, \sqrt{5} - 1\right), 1\right)} \]

       8. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       9. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

       10. lift--.f64 98.8

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   16. Applied rewrites 98.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(1 - \cos y\right)\right), \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            -0.0625
            (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* (cos (+ x x)) 0.5)))
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -6e-5)
     t_2
     (if (<= x 0.0135)
       (*
        (/
         (fma
          (* (- 0.5 (* (cos (+ y y)) 0.5)) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
        0.3333333333333333)
       t_2))))

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 = (fma(-0.0625, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6e-5) {
		tmp = t_2;
	} else if (x <= 0.0135) {
		tmp = (fma(((0.5 - (cos((y + y)) * 0.5)) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	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 = Float64(Float64(fma(-0.0625, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6e-5)
		tmp = t_2;
	elseif (x <= 0.0135)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(y + y)) * 0.5)) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0625 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6e-5], t$95$2, If[LessEqual[x, 0.0135], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000015e-5 or 0.0134999999999999998 < 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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 59.1%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -6.00000000000000015e-5 < x < 0.0134999999999999998

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} + \color{blue}{\frac{1}{24} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{24} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{3} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right) + \frac{1}{3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \sqrt{5} - \color{blue}{1}, \frac{1}{3}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \sqrt{5} - 1, \frac{1}{3}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right), \sqrt{5} - 1, \frac{1}{3}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right), \sqrt{5} - 1, \frac{1}{3}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right), \sqrt{5} - 1, \frac{1}{3}\right) \]

      8. lift--.f64 61.3

         \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \sqrt{5} - 1, 0.3333333333333333\right) \]

   7. Applied rewrites 61.3%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{\sqrt{5} - 1}, 0.3333333333333333\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   10. Applied rewrites 98.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(y + y\right) \cdot 0.5\right) \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 60.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    -0.0625
    (* (* (- (cos x) 1.0) (sqrt 2.0)) (- 0.5 (* (cos (+ x x)) 0.5)))
    2.0)
   (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma(-0.0625, (((cos(x) - 1.0) * sqrt(2.0)) * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(-0.0625, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(-0.0625 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

6. Applied rewrites 60.2%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 33: 45.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   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 / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / (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 / (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 / (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(2.0 / 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 / (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[(2.0 / 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]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

4. Applied rewrites 62.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

7. Applied rewrites 45.4%

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
8. Add Preprocessing

Alternative 34: 43.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/ 2.0 (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (2.0 / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 43.1%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 35: 40.6% accurate, 316.7× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.6%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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))))))