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

Percentage Accurate: 99.3% → 99.4%

Time: 15.8s

Alternatives: 41

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\\ \mathsf{fma}\left(0.3333333333333333, \frac{\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + t\_0}, 2 \cdot \frac{1}{3 - -3 \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (* (cos x) (- (sqrt 5.0) 1.0))
          (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))
   (fma
    0.3333333333333333
    (/
     (*
      (sqrt 2.0)
      (*
       (- (cos x) (cos y))
       (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* 0.0625 (sin x))))))
     (+ 1.0 t_0))
    (* 2.0 (/ 1.0 (- 3.0 (* -3.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), (2.0 * (cos(y) / (3.0 + sqrt(5.0)))));
	return fma(0.3333333333333333, ((sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x)))))) / (1.0 + t_0)), (2.0 * (1.0 / (3.0 - (-3.0 * t_0)))));
}

Julia

function code(x, y)
	t_0 = fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))
	return fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(0.0625 * sin(x)))))) / Float64(1.0 + t_0)), Float64(2.0 * Float64(1.0 / Float64(3.0 - Float64(-3.0 * t_0)))))
end

Wolfram

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

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 - \sqrt{5}}}}{2} \cdot \cos y\right)} \end{array} \]

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)))
    (*
     (/ (/ 4.0 (/ (- 9.0 (* (sqrt 5.0) (sqrt 5.0))) (- 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))) + (((4.0 / ((9.0 - (sqrt(5.0) * sqrt(5.0))) / (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))) + (((4.0d0 / ((9.0d0 - (sqrt(5.0d0) * sqrt(5.0d0))) / (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))) + (((4.0 / ((9.0 - (Math.sqrt(5.0) * Math.sqrt(5.0))) / (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))) + (((4.0 / ((9.0 - (math.sqrt(5.0) * math.sqrt(5.0))) / (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(4.0 / Float64(Float64(9.0 - Float64(sqrt(5.0) * sqrt(5.0))) / 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))) + (((4.0 / ((9.0 - (sqrt(5.0) * sqrt(5.0))) / (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[(4.0 / N[(N[(9.0 - N[(N[Sqrt[5.0], $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. 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%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 - \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\frac{9 - \color{blue}{\sqrt{5} \cdot \sqrt{5}}}{3 - \sqrt{5}}}}{2} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 - \color{blue}{\sqrt{5}}}}}{2} \cdot \cos y\right)} \]

   9. lift--.f64 99.3

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{\color{blue}{3 - \sqrt{5}}}}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 99.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{4}{\color{blue}{\frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 - \sqrt{5}}}}}{2} \cdot \cos y\right)} \]
6. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

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)))
    (* (/ (/ 4.0 (+ 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))) + (((4.0 / (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))) + (((4.0d0 / (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))) + (((4.0 / (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))) + (((4.0 / (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(4.0 / 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))) + (((4.0 / (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[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $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. 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%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

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))))
  (fma
   (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0)
   3.0
   (* (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) 3.0))))

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)))) / fma(fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0), 3.0, ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) * 3.0));
}

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)))) / fma(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0), 3.0, Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) * 3.0)))
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[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

3. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-sin.f64 99.3

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

6. Applied rewrites 99.3%

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

Alternative 7: 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 8: 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 9: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.85:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0005208333333333333 - 1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot y\right), -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_2
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          (*
           3.0
           (+
            (+ 1.0 (* t_1 (cos x)))
            (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))))
   (if (<= y -1e-7)
     t_2
     (if (<= y 0.85)
       (/
        (fma
         t_0
         (*
          (- (sin y) (/ (sin x) 16.0))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma
                (* y y)
                (- 0.0005208333333333333 (* 1.240079365079365e-5 (* y y)))
                -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (* (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0) (fma (cos x) t_1 1.0)) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double t_2 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((1.0 + (t_1 * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	double tmp;
	if (y <= -1e-7) {
		tmp = t_2;
	} else if (y <= 0.85) {
		tmp = fma(t_0, ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (y * (0.0625 + ((y * y) * fma((y * y), (0.0005208333333333333 - (1.240079365079365e-5 * (y * y))), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_1, 1.0)) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	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(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = t_2;
	elseif (y <= 0.85)
		tmp = Float64(fma(t_0, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(Float64(y * y), Float64(0.0005208333333333333 - Float64(1.240079365079365e-5 * Float64(y * y))), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_1, 1.0)) * 3.0));
	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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $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[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], t$95$2, If[LessEqual[y, 0.85], N[(N[(t$95$0 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 - N[(1.240079365079365e-5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999995e-8 or 0.849999999999999978 < y

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.3

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

   6. Applied rewrites 64.3%

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

   if -9.9999999999999995e-8 < y < 0.849999999999999978

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right)} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right)} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}\right) + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left({y}^{2}, \color{blue}{\frac{1}{1920} + \frac{-1}{80640} \cdot {y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{1920}} + \frac{-1}{80640} \cdot {y}^{2}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{1920}} + \frac{-1}{80640} \cdot {y}^{2}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{1920} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{80640}\right)\right) \cdot {y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{1920} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{80640}\right)\right) \cdot {y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{1920} - \left(\mathsf{neg}\left(\frac{-1}{80640}\right)\right) \cdot \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{1920} - \frac{1}{80640} \cdot {\color{blue}{y}}^{2}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{1920} - \frac{1}{80640} \cdot \left(y \cdot \color{blue}{y}\right), \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      16. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0005208333333333333 - 1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot \color{blue}{y}\right), -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.0005208333333333333 - 1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot y\right), -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_0 \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.49:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right), -0.5\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_1
         (/
          (+
           2.0
           (*
            (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
            (- (cos x) (cos y))))
          (*
           3.0
           (+
            (+ 1.0 (* t_0 (cos x)))
            (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))))
   (if (<= y -1e-7)
     t_1
     (if (<= y 0.49)
       (/
        (fma
         (-
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (fma
             (* y y)
             (- 0.041666666666666664 (* 0.001388888888888889 (* y y)))
             -0.5))))
         (*
          (- (sin y) (/ (sin x) 16.0))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (* (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0) (fma (cos x) t_0 1.0)) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_0 * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 0.49) {
		tmp = fma((cos(x) - (1.0 + ((y * y) * fma((y * y), (0.041666666666666664 - (0.001388888888888889 * (y * y))), -0.5)))), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_0, 1.0)) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = 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(3.0 * Float64(Float64(1.0 + Float64(t_0 * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 0.49)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 + Float64(Float64(y * y) * fma(Float64(y * y), Float64(0.041666666666666664 - Float64(0.001388888888888889 * Float64(y * y))), -0.5)))), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_0, 1.0)) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = 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[(3.0 * N[(N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], t$95$1, If[LessEqual[y, 0.49], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(0.041666666666666664 - N[(0.001388888888888889 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999995e-8 or 0.48999999999999999 < y

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.3

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

   6. Applied rewrites 64.3%

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

   if -9.9999999999999995e-8 < y < 0.48999999999999999

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) + \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left({y}^{2}, \color{blue}{\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{24}} + \frac{-1}{720} \cdot {y}^{2}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24} - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{720}\right)\right) \cdot {y}^{2}}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24} - \frac{1}{720} \cdot {\color{blue}{y}}^{2}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24} - \color{blue}{\frac{1}{720} \cdot {y}^{2}}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24} - \frac{1}{720} \cdot \color{blue}{{y}^{2}}, \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, \frac{1}{24} - \frac{1}{720} \cdot \left(y \cdot \color{blue}{y}\right), \frac{-1}{2}\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      15. lift-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot \color{blue}{y}\right), -0.5\right)\right), \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(y \cdot y, 0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right), -0.5\right)\right)}, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_0 \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_1
         (/
          (+
           2.0
           (*
            (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
            (- (cos x) (cos y))))
          (*
           3.0
           (+
            (+ 1.0 (* t_0 (cos x)))
            (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))))
   (if (<= y -1e-7)
     t_1
     (if (<= y 0.48)
       (/
        (fma
         (-
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (-
             (*
              (* y y)
              (- 0.041666666666666664 (* 0.001388888888888889 (* y y))))
             0.5))))
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (* (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0) (fma (cos x) t_0 1.0)) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_0 * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1;
	} else if (y <= 0.48) {
		tmp = fma((cos(x) - (1.0 + ((y * y) * (((y * y) * (0.041666666666666664 - (0.001388888888888889 * (y * y)))) - 0.5)))), (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_0, 1.0)) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = 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(3.0 * Float64(Float64(1.0 + Float64(t_0 * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = t_1;
	elseif (y <= 0.48)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * Float64(0.041666666666666664 - Float64(0.001388888888888889 * Float64(y * y)))) - 0.5)))), Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), t_0, 1.0)) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = 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[(3.0 * N[(N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], t$95$1, If[LessEqual[y, 0.48], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(0.041666666666666664 - N[(0.001388888888888889 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999995e-8 or 0.47999999999999998 < y

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.3

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

   6. Applied rewrites 64.3%

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

   if -9.9999999999999995e-8 < y < 0.47999999999999998

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \color{blue}{\frac{1}{2}}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \left(\mathsf{neg}\left(\frac{-1}{720}\right)\right) \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot \left(y \cdot y\right)\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       14. lift-*.f64 99.5

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (cos y)
           (/ (- 3.0 (sqrt 5.0)) 2.0)
           (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
          3.0))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
           2.0)
          t_0)))
   (if (<= y -0.28)
     t_1
     (if (<= y 0.48)
       (/
        (fma
         (-
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (-
             (*
              (* y y)
              (- 0.041666666666666664 (* 0.001388888888888889 (* y y))))
             0.5))))
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0;
	double t_1 = fma((cos(x) - cos(y)), (sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0;
	double tmp;
	if (y <= -0.28) {
		tmp = t_1;
	} else if (y <= 0.48) {
		tmp = fma((cos(x) - (1.0 + ((y * y) * (((y * y) * (0.041666666666666664 - (0.001388888888888889 * (y * y)))) - 0.5)))), (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.28)
		tmp = t_1;
	elseif (y <= 0.48)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * Float64(0.041666666666666664 - Float64(0.001388888888888889 * Float64(y * y)))) - 0.5)))), Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * 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[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.28], t$95$1, If[LessEqual[y, 0.48], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(0.041666666666666664 - N[(0.001388888888888889 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.28000000000000003 or 0.47999999999999998 < y

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if -0.28000000000000003 < y < 0.47999999999999998

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \color{blue}{\frac{1}{2}}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \left(\mathsf{neg}\left(\frac{-1}{720}\right)\right) \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot \left(y \cdot y\right)\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       14. lift-*.f64 99.4

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\\ t_3 := \frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_0, t\_2\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.48:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right), t\_0, t\_2\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
        (t_3
         (/
          (fma t_1 (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))) 2.0)
          (* (fma (cos y) t_0 t_2) 3.0))))
   (if (<= y -0.28)
     t_3
     (if (<= y 0.48)
       (/
        (fma
         t_1
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (*
         (fma
          (+
           1.0
           (*
            (* y y)
            (-
             (*
              (* y y)
              (- 0.041666666666666664 (* 0.001388888888888889 (* y y))))
             0.5)))
          t_0
          t_2)
         3.0))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0);
	double t_3 = fma(t_1, (sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (fma(cos(y), t_0, t_2) * 3.0);
	double tmp;
	if (y <= -0.28) {
		tmp = t_3;
	} else if (y <= 0.48) {
		tmp = fma(t_1, (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma((1.0 + ((y * y) * (((y * y) * (0.041666666666666664 - (0.001388888888888889 * (y * y)))) - 0.5))), t_0, t_2) * 3.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)
	t_3 = Float64(fma(t_1, Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), t_0, t_2) * 3.0))
	tmp = 0.0
	if (y <= -0.28)
		tmp = t_3;
	elseif (y <= 0.48)
		tmp = Float64(fma(t_1, Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * Float64(0.041666666666666664 - Float64(0.001388888888888889 * Float64(y * y)))) - 0.5))), t_0, t_2) * 3.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.28], t$95$3, If[LessEqual[y, 0.48], N[(N[(t$95$1 * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(0.041666666666666664 - N[(0.001388888888888889 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.28000000000000003 or 0.47999999999999998 < y

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if -0.28000000000000003 < y < 0.47999999999999998

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\color{blue}{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right)}, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right)} - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \color{blue}{\frac{1}{2}}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left({y}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \left(\mathsf{neg}\left(\frac{-1}{720}\right)\right) \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot {y}^{2}\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       13. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{24} - \frac{1}{720} \cdot \left(y \cdot y\right)\right) - \frac{1}{2}\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       14. lift-*.f64 99.4

           \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right), \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\color{blue}{1 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.041666666666666664 - 0.001388888888888889 \cdot \left(y \cdot y\right)\right) - 0.5\right)}, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.14:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 - 0.010416666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (cos y)
           (/ (- 3.0 (sqrt 5.0)) 2.0)
           (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
          3.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma t_1 (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))) 2.0)
          t_0)))
   (if (<= y -0.14)
     t_2
     (if (<= y 0.31)
       (/
        (fma
         t_1
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (- (sin x) (* y (- 0.0625 (* 0.010416666666666666 (* y y)))))
           (sqrt 2.0)))
         2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(t_1, (sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0;
	double tmp;
	if (y <= -0.14) {
		tmp = t_2;
	} else if (y <= 0.31) {
		tmp = fma(t_1, (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 - (0.010416666666666666 * (y * y))))) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(t_1, Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.14)
		tmp = t_2;
	elseif (y <= 0.31)
		tmp = Float64(fma(t_1, Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 - Float64(0.010416666666666666 * Float64(y * y))))) * sqrt(2.0))), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.14], t$95$2, If[LessEqual[y, 0.31], N[(N[(t$95$1 * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 - N[(0.010416666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 0.309999999999999998 < y

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if -0.14000000000000001 < y < 0.309999999999999998

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{\frac{-1}{96} \cdot {y}^{2}}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

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

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} - \left(\mathsf{neg}\left(\frac{-1}{96}\right)\right) \cdot \color{blue}{{y}^{2}}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} - \frac{1}{96} \cdot {y}^{2}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} - \frac{1}{96} \cdot \color{blue}{{y}^{2}}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} - \frac{1}{96} \cdot {y}^{\color{blue}{2}}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} - \frac{1}{96} \cdot \left(y \cdot y\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lift-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 - 0.010416666666666666 \cdot \left(y \cdot y\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 - \color{blue}{0.010416666666666666 \cdot \left(y \cdot y\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.28:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (cos y)
           (/ (- 3.0 (sqrt 5.0)) 2.0)
           (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
          3.0))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
           2.0)
          t_0)))
   (if (<= y -0.28)
     t_1
     (if (<= y 0.31)
       (/
        (fma
         (-
          (cos x)
          (+ 1.0 (* (* y y) (- (* 0.041666666666666664 (* y y)) 0.5))))
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0;
	double t_1 = fma((cos(x) - cos(y)), (sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0;
	double tmp;
	if (y <= -0.28) {
		tmp = t_1;
	} else if (y <= 0.31) {
		tmp = fma((cos(x) - (1.0 + ((y * y) * ((0.041666666666666664 * (y * y)) - 0.5)))), (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(sin(y) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.28)
		tmp = t_1;
	elseif (y <= 0.31)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5)))), Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * 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[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.28], t$95$1, If[LessEqual[y, 0.31], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.28000000000000003 or 0.309999999999999998 < y

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.2

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

   6. Applied rewrites 64.2%

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

   if -0.28000000000000003 < y < 0.309999999999999998

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \color{blue}{\frac{1}{2}}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := 1 + t\_1 \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_2 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.38:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_2 + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_2 (+ 1.0 (* t_1 (cos x)))))
   (if (<= y -1e-7)
     (/
      (+
       2.0
       (*
        (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_2 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 0.38)
       (/
        (fma
         (-
          (cos x)
          (+ 1.0 (* (* y y) (- (* 0.041666666666666664 (* y y)) 0.5))))
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (* (fma (cos y) t_0 (fma (cos x) t_1 1.0)) 3.0))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_2 (* t_0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double t_2 = 1.0 + (t_1 * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_2 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 0.38) {
		tmp = fma((cos(x) - (1.0 + ((y * y) * ((0.041666666666666664 * (y * y)) - 0.5)))), (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0);
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_2 + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_2 = Float64(1.0 + Float64(t_1 * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_2 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 0.38)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5)))), Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0));
	else
		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(t_2 + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.38], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], 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[(t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 0.38

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \color{blue}{\frac{1}{2}}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right), \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   if 0.38 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 17: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := 1 + t\_1 \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_2 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - -0.5 \cdot \left(y \cdot y\right)\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_2 + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_2 (+ 1.0 (* t_1 (cos x)))))
   (if (<= y -1e-7)
     (/
      (+
       2.0
       (*
        (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_2 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 0.25)
       (/
        (fma
         (- (- (cos x) (* -0.5 (* y y))) 1.0)
         (*
          (fma
           y
           (+
            1.0
            (*
             (* y y)
             (fma 0.008333333333333333 (* y y) -0.16666666666666666)))
           (* -0.0625 (sin x)))
          (*
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (fma 0.0005208333333333333 (* y y) -0.010416666666666666)))))
           (sqrt 2.0)))
         2.0)
        (* (fma (cos y) t_0 (fma (cos x) t_1 1.0)) 3.0))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_2 (* t_0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double t_2 = 1.0 + (t_1 * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_2 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 0.25) {
		tmp = fma(((cos(x) - (-0.5 * (y * y))) - 1.0), (fma(y, (1.0 + ((y * y) * fma(0.008333333333333333, (y * y), -0.16666666666666666))), (-0.0625 * sin(x))) * ((sin(x) - (y * (0.0625 + ((y * y) * fma(0.0005208333333333333, (y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0);
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_2 + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_2 = Float64(1.0 + Float64(t_1 * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_2 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 0.25)
		tmp = Float64(fma(Float64(Float64(cos(x) - Float64(-0.5 * Float64(y * y))) - 1.0), Float64(fma(y, Float64(1.0 + Float64(Float64(y * y) * fma(0.008333333333333333, Float64(y * y), -0.16666666666666666))), Float64(-0.0625 * sin(x))) * Float64(Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * fma(0.0005208333333333333, Float64(y * y), -0.010416666666666666))))) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0));
	else
		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(t_2 + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.25], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 * N[(y * y), $MachinePrecision] + -0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], 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[(t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 0.25

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \color{blue}{\left(\frac{1}{16} + {y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \color{blue}{{y}^{2} \cdot \left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + {y}^{2} \cdot \color{blue}{\left(\frac{1}{1920} \cdot {y}^{2} - \frac{1}{96}\right)}\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{1920} \cdot {y}^{2}} - \frac{1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{96}\right)\right)}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \left(\frac{1}{1920} \cdot {y}^{2} + \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, \color{blue}{{y}^{2}}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot \color{blue}{y}, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-*.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot \color{blue}{y}, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \color{blue}{y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \frac{-1}{16} \cdot \sin \color{blue}{x}\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)}, \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {y}^{2}} - \frac{1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. negate-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{{y}^{2}}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot \color{blue}{y}, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      14. lift-sin.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   9. Applied rewrites 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right)} \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\cos x + \frac{1}{2} \cdot {y}^{2}\right) - 1}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x + \frac{1}{2} \cdot {y}^{2}\right) - \color{blue}{1}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot {y}^{2}\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \frac{-1}{2} \cdot {y}^{2}\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       4. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \frac{-1}{2} \cdot {y}^{2}\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \frac{-1}{2} \cdot {y}^{2}\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \frac{-1}{2} \cdot {y}^{2}\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       7. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \frac{-1}{2} \cdot \left(y \cdot y\right)\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{-1}{6}\right), \frac{-1}{16} \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(\frac{1}{16} + \left(y \cdot y\right) \cdot \mathsf{fma}\left(\frac{1}{1920}, y \cdot y, \frac{-1}{96}\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

       8. lift-*.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - -0.5 \cdot \left(y \cdot y\right)\right) - 1, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   12. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\cos x - -0.5 \cdot \left(y \cdot y\right)\right) - 1}, \mathsf{fma}\left(y, 1 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \mathsf{fma}\left(0.0005208333333333333, y \cdot y, -0.010416666666666666\right)\right)\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   if 0.25 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 18: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5} - 1}{2}\\ t_1 := 1 + t\_0 \cdot \cos x\\ t_2 := \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_1 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \left(1 - 0.5 \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(1.00390625 \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_2, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_1 + t\_2 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_1 (+ 1.0 (* t_0 (cos x))))
        (t_2 (/ (- 3.0 (sqrt 5.0)) 2.0)))
   (if (<= y -1e-7)
     (/
      (+
       2.0
       (*
        (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_1 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 0.25)
       (/
        (fma
         (- (cos x) (- 1.0 (* 0.5 (* y y))))
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
          (sqrt 2.0)
          (*
           y
           (fma
            (* -0.0625 y)
            (sqrt 2.0)
            (* (sqrt 2.0) (* 1.00390625 (sin x))))))
         2.0)
        (* (fma (cos y) t_2 (fma (cos x) t_0 1.0)) 3.0))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_1 (* t_2 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) / 2.0;
	double t_1 = 1.0 + (t_0 * cos(x));
	double t_2 = (3.0 - sqrt(5.0)) / 2.0;
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_1 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 0.25) {
		tmp = fma((cos(x) - (1.0 - (0.5 * (y * y)))), fma((-0.0625 * (0.5 - (0.5 * cos((x + x))))), sqrt(2.0), (y * fma((-0.0625 * y), sqrt(2.0), (sqrt(2.0) * (1.00390625 * sin(x)))))), 2.0) / (fma(cos(y), t_2, fma(cos(x), t_0, 1.0)) * 3.0);
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_1 + (t_2 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_1 = Float64(1.0 + Float64(t_0 * cos(x)))
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_1 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 0.25)
		tmp = Float64(fma(Float64(cos(x) - Float64(1.0 - Float64(0.5 * Float64(y * y)))), fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), sqrt(2.0), Float64(y * fma(Float64(-0.0625 * y), sqrt(2.0), Float64(sqrt(2.0) * Float64(1.00390625 * sin(x)))))), 2.0) / Float64(fma(cos(y), t_2, fma(cos(x), t_0, 1.0)) * 3.0));
	else
		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(t_1 + Float64(t_2 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.25], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[(1.0 - N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(y * N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], 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[(t$95$1 + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 0.25

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{\left(1 + \frac{-1}{2} \cdot {y}^{2}\right)}, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

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

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {y}^{2}}\right), \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot {\color{blue}{y}}^{2}\right), \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \color{blue}{\frac{1}{2} \cdot {y}^{2}}\right), \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot \color{blue}{{y}^{2}}\right), \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \left(1 - \frac{1}{2} \cdot \left(y \cdot \color{blue}{y}\right)\right), \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. lift-*.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if 0.25 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 19: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := 1 + t\_1 \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_2 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - 1, \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(1.00390625 \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_2 + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_2 (+ 1.0 (* t_1 (cos x)))))
   (if (<= y -1e-7)
     (/
      (+
       2.0
       (*
        (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_2 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 0.25)
       (/
        (fma
         (- (cos x) 1.0)
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
          (sqrt 2.0)
          (*
           y
           (fma
            (* -0.0625 y)
            (sqrt 2.0)
            (* (sqrt 2.0) (* 1.00390625 (sin x))))))
         2.0)
        (* (fma (cos y) t_0 (fma (cos x) t_1 1.0)) 3.0))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_2 (* t_0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double t_2 = 1.0 + (t_1 * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_2 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 0.25) {
		tmp = fma((cos(x) - 1.0), fma((-0.0625 * (0.5 - (0.5 * cos((x + x))))), sqrt(2.0), (y * fma((-0.0625 * y), sqrt(2.0), (sqrt(2.0) * (1.00390625 * sin(x)))))), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0);
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_2 + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_2 = Float64(1.0 + Float64(t_1 * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_2 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 0.25)
		tmp = Float64(fma(Float64(cos(x) - 1.0), fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), sqrt(2.0), Float64(y * fma(Float64(-0.0625 * y), sqrt(2.0), Float64(sqrt(2.0) * Float64(1.00390625 * sin(x)))))), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0));
	else
		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(t_2 + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.25], N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(y * N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], 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[(t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 0.25

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \color{blue}{1}, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

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

   if 0.25 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 20: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := 1 + \frac{t\_2}{2} \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \left(t\_3 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(-0.0625 \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(1.00390625 \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot t\_1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_3 + \frac{t\_1}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (+ 1.0 (* (/ t_2 2.0) (cos x)))))
   (if (<= y -1e-7)
     (/
      (+ 2.0 (* (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0)) t_0))
      (* 3.0 (+ t_3 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))
     (if (<= y 3.8e-5)
       (/
        (fma
         t_0
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
          (sqrt 2.0)
          (*
           y
           (fma
            (* -0.0625 y)
            (sqrt 2.0)
            (* (sqrt 2.0) (* 1.00390625 (sin x))))))
         2.0)
        (* (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 t_1))) 3.0))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_3 (* (/ t_1 2.0) (cos y)))))))))

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 = sqrt(5.0) - 1.0;
	double t_3 = 1.0 + ((t_2 / 2.0) * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * t_0)) / (3.0 * (t_3 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	} else if (y <= 3.8e-5) {
		tmp = fma(t_0, fma((-0.0625 * (0.5 - (0.5 * cos((x + x))))), sqrt(2.0), (y * fma((-0.0625 * y), sqrt(2.0), (sqrt(2.0) * (1.00390625 * sin(x)))))), 2.0) / ((1.0 + fma(0.5, (cos(x) * t_2), (0.5 * t_1))) * 3.0);
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_3 + ((t_1 / 2.0) * cos(y))));
	}
	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(sqrt(5.0) - 1.0)
	t_3 = Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * t_0)) / Float64(3.0 * Float64(t_3 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	elseif (y <= 3.8e-5)
		tmp = Float64(fma(t_0, fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), sqrt(2.0), Float64(y * fma(Float64(-0.0625 * y), sqrt(2.0), Float64(sqrt(2.0) * Float64(1.00390625 * sin(x)))))), 2.0) / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * t_1))) * 3.0));
	else
		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(t_3 + Float64(Float64(t_1 / 2.0) * cos(y)))));
	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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-5], N[(N[(t$95$0 * N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(y * N[(N[(-0.0625 * y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], 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[(t$95$3 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 3.8000000000000002e-5

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot 3} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right) \cdot 3} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \color{blue}{\left(\sqrt{5} - 1\right)}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\color{blue}{\sqrt{5}} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - \color{blue}{1}\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right), \sqrt{2}, y \cdot \mathsf{fma}\left(\frac{-1}{16} \cdot y, \sqrt{2}, \sqrt{2} \cdot \left(\frac{257}{256} \cdot \sin x\right)\right)\right), 2\right)}{\left(1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3} \]

      9. lift--.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 3.8000000000000002e-5 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 21: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ t_3 := \sqrt{5} - 1\\ t_4 := \frac{t\_3}{2}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(-0.0625 \cdot t\_2\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_1}{2}, \mathsf{fma}\left(\cos x, t\_4, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(-0.0625 \cdot x, \sqrt{2}, \left(1.00390625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot t\_0}{3 + \left(0.5 \cdot \mathsf{fma}\left(\cos y, t\_1, t\_3\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + t\_4 \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 0.5 (* 0.5 (cos (+ x x)))))
        (t_3 (- (sqrt 5.0) 1.0))
        (t_4 (/ t_3 2.0)))
   (if (<= x -1.05e-5)
     (/
      (fma t_0 (* (* -0.0625 t_2) (sqrt 2.0)) 2.0)
      (* (fma (cos y) (/ t_1 2.0) (fma (cos x) t_4 1.0)) 3.0))
     (if (<= x 9e-7)
       (/
        (+
         2.0
         (*
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (sqrt 2.0)
           (*
            (fma
             (* -0.0625 x)
             (sqrt 2.0)
             (* (* 1.00390625 (sin y)) (sqrt 2.0)))
            x))
          t_0))
        (+ 3.0 (* (* 0.5 (fma (cos y) t_1 t_3)) 3.0)))
       (/
        (- 2.0 (* 0.0625 (* t_2 (* (sqrt 2.0) (- (cos x) 1.0)))))
        (*
         3.0
         (+
          (+ 1.0 (* t_4 (cos x)))
          (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))))))

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 = 0.5 - (0.5 * cos((x + x)));
	double t_3 = sqrt(5.0) - 1.0;
	double t_4 = t_3 / 2.0;
	double tmp;
	if (x <= -1.05e-5) {
		tmp = fma(t_0, ((-0.0625 * t_2) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_1 / 2.0), fma(cos(x), t_4, 1.0)) * 3.0);
	} else if (x <= 9e-7) {
		tmp = (2.0 + (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), sqrt(2.0), (fma((-0.0625 * x), sqrt(2.0), ((1.00390625 * sin(y)) * sqrt(2.0))) * x)) * t_0)) / (3.0 + ((0.5 * fma(cos(y), t_1, t_3)) * 3.0));
	} else {
		tmp = (2.0 - (0.0625 * (t_2 * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 * ((1.0 + (t_4 * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	}
	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(0.5 - Float64(0.5 * cos(Float64(x + x))))
	t_3 = Float64(sqrt(5.0) - 1.0)
	t_4 = Float64(t_3 / 2.0)
	tmp = 0.0
	if (x <= -1.05e-5)
		tmp = Float64(fma(t_0, Float64(Float64(-0.0625 * t_2) * sqrt(2.0)), 2.0) / Float64(fma(cos(y), Float64(t_1 / 2.0), fma(cos(x), t_4, 1.0)) * 3.0));
	elseif (x <= 9e-7)
		tmp = Float64(Float64(2.0 + Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), sqrt(2.0), Float64(fma(Float64(-0.0625 * x), sqrt(2.0), Float64(Float64(1.00390625 * sin(y)) * sqrt(2.0))) * x)) * t_0)) / Float64(3.0 + Float64(Float64(0.5 * fma(cos(y), t_1, t_3)) * 3.0)));
	else
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_4 * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))));
	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[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $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]}, If[LessEqual[x, -1.05e-5], N[(N[(t$95$0 * N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(N[(-0.0625 * x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$4 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04999999999999994e-5

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 61.2

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

   6. Applied rewrites 61.2%

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

   if -1.04999999999999994e-5 < x < 8.99999999999999959e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{x} \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16} \cdot x, \sqrt{2}, \left(\frac{257}{256} \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16} \cdot x, \sqrt{2}, \left(\frac{257}{256} \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot \left(\cos x - \cos y\right)}{1 \cdot 3 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot 3}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16} \cdot x, \sqrt{2}, \left(\frac{257}{256} \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot 3} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16} \cdot x, \sqrt{2}, \left(\frac{257}{256} \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot 3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right), \sqrt{2}, \mathsf{fma}\left(\frac{-1}{16} \cdot x, \sqrt{2}, \left(\frac{257}{256} \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot x\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{3}} \]

   7. Applied rewrites 99.5%

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

   if 8.99999999999999959e-7 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.4%

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

Alternative 22: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{t\_1}{2}\\ t_3 := 1 + t\_2 \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(t\_3 + \frac{\frac{4}{t\_0}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(t\_2, \cos x, \frac{2}{t\_0}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 2 \cdot \frac{1}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_3 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (/ t_1 2.0))
        (t_3 (+ 1.0 (* t_2 (cos x)))))
   (if (<= y -1e-7)
     (/
      (+
       2.0
       (*
        (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ y y))))) (sqrt 2.0))
        (- (cos x) (cos y))))
      (* 3.0 (+ t_3 (* (/ (/ 4.0 t_0) 2.0) (cos y)))))
     (if (<= y 7.3e-6)
       (+
        (/ 2.0 (* 3.0 (+ 1.0 (fma t_2 (cos x) (/ 2.0 t_0)))))
        (*
         -0.020833333333333332
         (/
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))
          (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 2.0 (/ 1.0 t_0)))))))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_3 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))))

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 = t_1 / 2.0;
	double t_3 = 1.0 + (t_2 * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((y + y))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * (t_3 + (((4.0 / t_0) / 2.0) * cos(y))));
	} else if (y <= 7.3e-6) {
		tmp = (2.0 / (3.0 * (1.0 + fma(t_2, cos(x), (2.0 / t_0))))) + (-0.020833333333333332 * (((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 + fma(0.5, (cos(x) * t_1), (2.0 * (1.0 / t_0))))));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_3 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	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(t_1 / 2.0)
	t_3 = Float64(1.0 + Float64(t_2 * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(y + y))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(t_3 + Float64(Float64(Float64(4.0 / t_0) / 2.0) * cos(y)))));
	elseif (y <= 7.3e-6)
		tmp = Float64(Float64(2.0 / Float64(3.0 * Float64(1.0 + fma(t_2, cos(x), Float64(2.0 / t_0))))) + Float64(-0.020833333333333332 * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(2.0 * Float64(1.0 / t_0)))))));
	else
		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(t_3 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	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[(t$95$1 / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(N[(4.0 / t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.3e-6], N[(N[(2.0 / N[(3.0 * N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(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] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(t$95$3 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

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

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift-sqrt.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -9.9999999999999995e-8 < y < 7.30000000000000041e-6

   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 99.6

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

   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 99.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{\frac{-1}{48} \cdot \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{-0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{3 + \sqrt{5}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       3. lift-+.f64 99.1

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{3 + \color{blue}{\sqrt{5}}}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   if 7.30000000000000041e-6 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 23: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := 3 + \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_2}{2}\\ t_4 := 1 + t\_3 \cdot \cos x\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 \cdot \left(t\_4 + \frac{\frac{4}{t\_1}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(t\_3, \cos x, \frac{2}{t\_1}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 2 \cdot \frac{1}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_4 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_2 2.0))
        (t_4 (+ 1.0 (* t_3 (cos x)))))
   (if (<= y -1e-7)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) t_0))))
      (* 3.0 (+ t_4 (* (/ (/ 4.0 t_1) 2.0) (cos y)))))
     (if (<= y 7.3e-6)
       (+
        (/ 2.0 (* 3.0 (+ 1.0 (fma t_3 (cos x) (/ 2.0 t_1)))))
        (*
         -0.020833333333333332
         (/
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))
          (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 2.0 (/ 1.0 t_1)))))))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* t_0 (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_4 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = 3.0 + sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_2 / 2.0;
	double t_4 = 1.0 + (t_3 * cos(x));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * t_0)))) / (3.0 * (t_4 + (((4.0 / t_1) / 2.0) * cos(y))));
	} else if (y <= 7.3e-6) {
		tmp = (2.0 / (3.0 * (1.0 + fma(t_3, cos(x), (2.0 / t_1))))) + (-0.020833333333333332 * (((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 + fma(0.5, (cos(x) * t_2), (2.0 * (1.0 / t_1))))));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_0 * sqrt(2.0)), 2.0) / (3.0 * (t_4 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(3.0 + sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 / 2.0)
	t_4 = Float64(1.0 + Float64(t_3 * cos(x)))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 * Float64(t_4 + Float64(Float64(Float64(4.0 / t_1) / 2.0) * cos(y)))));
	elseif (y <= 7.3e-6)
		tmp = Float64(Float64(2.0 / Float64(3.0 * Float64(1.0 + fma(t_3, cos(x), Float64(2.0 / t_1))))) + Float64(-0.020833333333333332 * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(2.0 * Float64(1.0 / t_1)))))));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_4 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $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[(t$95$2 / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(N[(N[(4.0 / t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.3e-6], N[(N[(2.0 / N[(3.0 * N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(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] / N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$4 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.2%

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

   if -9.9999999999999995e-8 < y < 7.30000000000000041e-6

   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 99.6

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

   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 99.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{\frac{-1}{48} \cdot \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{-0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{3 + \sqrt{5}}\right)\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       3. lift-+.f64 99.1

          \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{2}{3 + \color{blue}{\sqrt{5}}}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)\right)} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   if 7.30000000000000041e-6 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 24: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := 3 + \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := 1 + \frac{t\_2}{2} \cdot \cos x\\ t_4 := 1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 2 \cdot \frac{1}{t\_1}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 \cdot \left(t\_3 + \frac{\frac{4}{t\_1}}{2} \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.6666666666666666}{t\_4} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_3 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (+ 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (+ 1.0 (* (/ t_2 2.0) (cos x))))
        (t_4 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 2.0 (/ 1.0 t_1))))))
   (if (<= y -1e-7)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) t_0))))
      (* 3.0 (+ t_3 (* (/ (/ 4.0 t_1) 2.0) (cos y)))))
     (if (<= y 7.3e-6)
       (+
        (/ 0.6666666666666666 t_4)
        (*
         -0.020833333333333332
         (/
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))
          t_4)))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* t_0 (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_3 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = 3.0 + sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 1.0 + ((t_2 / 2.0) * cos(x));
	double t_4 = 1.0 + fma(0.5, (cos(x) * t_2), (2.0 * (1.0 / t_1)));
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * t_0)))) / (3.0 * (t_3 + (((4.0 / t_1) / 2.0) * cos(y))));
	} else if (y <= 7.3e-6) {
		tmp = (0.6666666666666666 / t_4) + (-0.020833333333333332 * (((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))) / t_4));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_0 * sqrt(2.0)), 2.0) / (3.0 * (t_3 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(3.0 + sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x)))
	t_4 = Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(2.0 * Float64(1.0 / t_1))))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 * Float64(t_3 + Float64(Float64(Float64(4.0 / t_1) / 2.0) * cos(y)))));
	elseif (y <= 7.3e-6)
		tmp = Float64(Float64(0.6666666666666666 / t_4) + Float64(-0.020833333333333332 * Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / t_4)));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_3 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $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[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(N[(4.0 / t$95$1), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.3e-6], N[(N[(0.6666666666666666 / t$95$4), $MachinePrecision] + N[(-0.020833333333333332 * N[(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] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.2%

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

   if -9.9999999999999995e-8 < y < 7.30000000000000041e-6

   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 99.6

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

   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 99.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{\frac{-1}{48} \cdot \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{2}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\right)} + \color{blue}{-0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       2. lift-cos.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       4. lift--.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       7. lift-+.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \color{blue}{\sqrt{5}}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{\color{blue}{3 + \sqrt{5}}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \color{blue}{\frac{1}{3 + \sqrt{5}}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{\frac{2}{3}}{1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}, 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} + \frac{-1}{48} \cdot \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\frac{0.6666666666666666}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}} + -0.020833333333333332 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 + \mathsf{fma}\left(0.5, \cos x \cdot \left(\sqrt{5} - 1\right), 2 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

   if 7.30000000000000041e-6 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 25: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_2 := 3 \cdot \left(t\_1 + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)\\ \mathbf{if}\;y \leq -0.00165:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{t\_2}\\ \mathbf{elif}\;y \leq 0.25:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_2 (* 3.0 (+ t_1 (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y))))))
   (if (<= y -0.00165)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) t_0))))
      t_2)
     (if (<= y 0.25)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        t_2)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* t_0 (sqrt 2.0))
         2.0)
        (* 3.0 (+ t_1 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_2 = 3.0 * (t_1 + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y)));
	double tmp;
	if (y <= -0.00165) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * t_0)))) / t_2;
	} else if (y <= 0.25) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_2;
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_0 * sqrt(2.0)), 2.0) / (3.0 * (t_1 + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_2 = Float64(3.0 * Float64(t_1 + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y))))
	tmp = 0.0
	if (y <= -0.00165)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * t_0)))) / t_2);
	elseif (y <= 0.25)
		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))))) / t_2);
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_0 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_1 + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(t$95$1 + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00165], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 0.25], 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] / t$95$2), $MachinePrecision], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00165

   1. Initial program 99.1%

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

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.1%

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

   if -0.00165 < y < 0.25

   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%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.6%

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

   if 0.25 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 26: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{\frac{4}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -0.00032:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot t\_0\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_0, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (-
           2.0
           (*
            0.0625
            (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (*
           3.0
           (+
            (+ 1.0 (* (/ t_0 2.0) (cos x)))
            (* (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0) (cos y)))))))
   (if (<= x -0.00032)
     t_1
     (if (<= x 9e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma
         3.0
         (+ 1.0 (fma 0.5 (* (cos y) (- 3.0 (sqrt 5.0))) (* 0.5 t_0)))
         (* (* x x) (fma -0.75 t_0 (* 0.0625 (* (* x x) t_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))))) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((4.0 / (3.0 + sqrt(5.0))) / 2.0) * cos(y))));
	double tmp;
	if (x <= -0.00032) {
		tmp = t_1;
	} else if (x <= 9e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(3.0, (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * t_0))), ((x * x) * fma(-0.75, t_0, (0.0625 * ((x * x) * 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(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(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0) * cos(y)))))
	tmp = 0.0
	if (x <= -0.00032)
		tmp = t_1;
	elseif (x <= 9e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(3.0, Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * t_0))), Float64(Float64(x * x) * fma(-0.75, t_0, Float64(0.0625 * Float64(Float64(x * x) * 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[(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[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00032], t$95$1, If[LessEqual[x, 9e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$0 + N[(0.0625 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000026e-4 or 8.99999999999999959e-7 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.8%

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

   if -3.20000000000000026e-4 < x < 8.99999999999999959e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.1%

      \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{t\_0}{2}\\ t_2 := 1 - \cos y\\ t_3 := \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot t\_2\right)\right)}{\mathsf{fma}\left(\cos y, t\_3, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_2 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_3 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ t_0 2.0))
        (t_2 (- 1.0 (cos y)))
        (t_3 (/ (- 3.0 (sqrt 5.0)) 2.0)))
   (if (<= y -1e-7)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) t_2))))
      (* (fma (cos y) t_3 (fma (cos x) t_1 1.0)) 3.0))
     (if (<= y 7.3e-6)
       (*
        0.3333333333333333
        (/
         (-
          2.0
          (*
           0.0625
           (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+ 1.0 (fma (* 0.5 (cos x)) t_0 (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* t_2 (sqrt 2.0))
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_3 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = t_0 / 2.0;
	double t_2 = 1.0 - cos(y);
	double t_3 = (3.0 - sqrt(5.0)) / 2.0;
	double tmp;
	if (y <= -1e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * t_2)))) / (fma(cos(y), t_3, fma(cos(x), t_1, 1.0)) * 3.0);
	} else if (y <= 7.3e-6) {
		tmp = 0.3333333333333333 * ((2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + fma((0.5 * cos(x)), t_0, (2.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_2 * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_3 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(1.0 - cos(y))
	t_3 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * t_2)))) / Float64(fma(cos(y), t_3, fma(cos(x), t_1, 1.0)) * 3.0));
	elseif (y <= 7.3e-6)
		tmp = Float64(0.3333333333333333 * 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(1.0 + fma(Float64(0.5 * cos(x)), t_0, Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(t_2 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_3 * cos(y)))));
	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[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$3 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.3e-6], N[(0.3333333333333333 * 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[(1.0 + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.2%

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

   if -9.9999999999999995e-8 < y < 7.30000000000000041e-6

   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 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 99.0%

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

   if 7.30000000000000041e-6 < y

   1. Initial program 99.1%

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

   2. 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 61.3%

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

Alternative 28: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot \left(\cos y \cdot t\_2\right)\right)\right) \cdot 3}\\ \end{array} \end{array} \]

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 (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y)))))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -1e-7)
     (/ t_1 (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (if (<= y 7.3e-6)
       (*
        0.3333333333333333
        (/
         (-
          2.0
          (*
           0.0625
           (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+ 1.0 (fma (* 0.5 (cos x)) t_0 (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
       (/
        t_1
        (* (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 (* (cos y) t_2)))) 3.0))))))

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((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -1e-7) {
		tmp = t_1 / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else if (y <= 7.3e-6) {
		tmp = 0.3333333333333333 * ((2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + fma((0.5 * cos(x)), t_0, (2.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = t_1 / ((1.0 + fma(0.5, (cos(x) * t_0), (0.5 * (cos(y) * t_2)))) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -1e-7)
		tmp = Float64(t_1 / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	elseif (y <= 7.3e-6)
		tmp = Float64(0.3333333333333333 * 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(1.0 + fma(Float64(0.5 * cos(x)), t_0, Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(t_1 / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * Float64(cos(y) * t_2)))) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-7], N[(t$95$1 / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.3e-6], N[(0.3333333333333333 * 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[(1.0 + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999995e-8

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.2%

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

   if -9.9999999999999995e-8 < y < 7.30000000000000041e-6

   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 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 99.0%

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

   if 7.30000000000000041e-6 < y

   1. Initial program 99.1%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 61.3%

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

   7. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 61.3

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

   9. Applied rewrites 61.3%

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

Alternative 29: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \cos y \cdot t\_2\\ \mathbf{if}\;x \leq -0.00032:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, t\_3, 0.5 \cdot t\_0\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_0, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_3\right)\right) \cdot 3}\\ \end{array} \end{array} \]

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))))))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* (cos y) t_2)))
   (if (<= x -0.00032)
     (/ t_1 (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
     (if (<= x 9e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma
         3.0
         (+ 1.0 (fma 0.5 t_3 (* 0.5 t_0)))
         (* (* x x) (fma -0.75 t_0 (* 0.0625 (* (* x x) t_0))))))
       (/ t_1 (* (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_3))) 3.0))))))

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))));
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = cos(y) * t_2;
	double tmp;
	if (x <= -0.00032) {
		tmp = t_1 / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0);
	} else if (x <= 9e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(3.0, (1.0 + fma(0.5, t_3, (0.5 * t_0))), ((x * x) * fma(-0.75, t_0, (0.0625 * ((x * x) * t_0)))));
	} else {
		tmp = t_1 / ((1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_3))) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 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(3.0 - sqrt(5.0))
	t_3 = Float64(cos(y) * t_2)
	tmp = 0.0
	if (x <= -0.00032)
		tmp = Float64(t_1 / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0));
	elseif (x <= 9e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(3.0, Float64(1.0 + fma(0.5, t_3, Float64(0.5 * t_0))), Float64(Float64(x * x) * fma(-0.75, t_0, Float64(0.0625 * Float64(Float64(x * x) * t_0))))));
	else
		tmp = Float64(t_1 / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_3))) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, -0.00032], N[(t$95$1 / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * t$95$3 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$0 + N[(0.0625 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000026e-4

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 61.2%

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

   if -3.20000000000000026e-4 < x < 8.99999999999999959e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.1%

      \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   if 8.99999999999999959e-7 < x

   1. Initial program 99.0%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 60.4

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

   9. Applied rewrites 60.4%

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

Alternative 30: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos y \cdot \left(3 - \sqrt{5}\right)\\ t_2 := \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_0, 0.5 \cdot t\_1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.00032:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, t\_1, 0.5 \cdot t\_0\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_0, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (cos y) (- 3.0 (sqrt 5.0))))
        (t_2
         (/
          (-
           2.0
           (*
            0.0625
            (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (* (+ 1.0 (fma 0.5 (* (cos x) t_0) (* 0.5 t_1))) 3.0))))
   (if (<= x -0.00032)
     t_2
     (if (<= x 9e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma
         3.0
         (+ 1.0 (fma 0.5 t_1 (* 0.5 t_0)))
         (* (* x x) (fma -0.75 t_0 (* 0.0625 (* (* x x) t_0))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(y) * (3.0 - sqrt(5.0));
	double t_2 = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / ((1.0 + fma(0.5, (cos(x) * t_0), (0.5 * t_1))) * 3.0);
	double tmp;
	if (x <= -0.00032) {
		tmp = t_2;
	} else if (x <= 9e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(3.0, (1.0 + fma(0.5, t_1, (0.5 * t_0))), ((x * x) * fma(-0.75, t_0, (0.0625 * ((x * x) * t_0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(y) * Float64(3.0 - sqrt(5.0)))
	t_2 = 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(Float64(1.0 + fma(0.5, Float64(cos(x) * t_0), Float64(0.5 * t_1))) * 3.0))
	tmp = 0.0
	if (x <= -0.00032)
		tmp = t_2;
	elseif (x <= 9e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(3.0, Float64(1.0 + fma(0.5, t_1, Float64(0.5 * t_0))), Float64(Float64(x * x) * fma(-0.75, t_0, Float64(0.0625 * Float64(Float64(x * x) * t_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[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00032], t$95$2, If[LessEqual[x, 9e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * t$95$1 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$0 + N[(0.0625 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000026e-4 or 8.99999999999999959e-7 < x

   1. Initial program 99.0%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 60.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 60.8

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

   9. Applied rewrites 60.8%

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

   if -3.20000000000000026e-4 < x < 8.99999999999999959e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 99.1%

      \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 79.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.125:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{1 + \mathsf{fma}\left(0.5 \cdot \cos x, t\_1, \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 0.00068:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot t\_1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.125)
     (*
      0.3333333333333333
      (/
       (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0))))
       (+ 1.0 (fma (* 0.5 (cos x)) t_1 (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 0.00068)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma
         3.0
         (+ 1.0 (fma 0.5 (* (cos y) (- 3.0 (sqrt 5.0))) (* 0.5 t_1)))
         (* (* x x) (fma -0.75 t_1 (* 0.0625 (* (* x x) t_1))))))
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.125) {
		tmp = 0.3333333333333333 * ((2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_0)))) / (1.0 + fma((0.5 * cos(x)), t_1, (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 0.00068) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(3.0, (1.0 + fma(0.5, (cos(y) * (3.0 - sqrt(5.0))), (0.5 * t_1))), ((x * x) * fma(-0.75, t_1, (0.0625 * ((x * x) * t_1)))));
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.125)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_0)))) / Float64(1.0 + fma(Float64(0.5 * cos(x)), t_1, Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 0.00068)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(3.0, Float64(1.0 + fma(0.5, Float64(cos(y) * Float64(3.0 - sqrt(5.0))), Float64(0.5 * t_1))), Float64(Float64(x * x) * fma(-0.75, t_1, Float64(0.0625 * Float64(Float64(x * x) * t_1))))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.125], N[(0.3333333333333333 * 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00068], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$1 + N[(0.0625 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $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] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.125

   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 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 60.2%

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

   if -0.125 < x < 6.8e-4

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 98.7%

      \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   if 6.8e-4 < x

   1. Initial program 99.0%

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

   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 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. 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. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 59.3

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

   6. Applied rewrites 59.3%

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

Alternative 32: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{1 + \mathsf{fma}\left(0.5 \cdot \cos x, t\_1, \frac{2}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, 1 - -0.5 \cdot t\_1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0)) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<= x -4.6e-6)
     (*
      0.3333333333333333
      (/
       (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0))))
       (+ 1.0 (fma (* 0.5 (cos x)) t_1 (/ 2.0 (+ 3.0 (sqrt 5.0)))))))
     (if (<= x 7.7e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* (fma (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0) (- 1.0 (* -0.5 t_1))) 3.0))
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -4.6e-6) {
		tmp = 0.3333333333333333 * ((2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_0)))) / (1.0 + fma((0.5 * cos(x)), t_1, (2.0 / (3.0 + sqrt(5.0))))));
	} else if (x <= 7.7e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / (fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), (1.0 - (-0.5 * t_1))) * 3.0);
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -4.6e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_0)))) / Float64(1.0 + fma(Float64(0.5 * cos(x)), t_1, Float64(2.0 / Float64(3.0 + sqrt(5.0)))))));
	elseif (x <= 7.7e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), Float64(1.0 - Float64(-0.5 * t_1))) * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -4.6e-6], N[(0.3333333333333333 * 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(2.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.7e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(1.0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $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$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-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 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 60.1%

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

   if -4.6e-6 < x < 7.7000000000000004e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if 7.7000000000000004e-7 < x

   1. Initial program 99.0%

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

   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 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   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. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 33: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot t\_2\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, 1 - -0.5 \cdot t\_1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -4.6e-6)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0))))
      (* (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 t_2))) 3.0))
     (if (<= x 7.7e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* (fma (cos y) (/ t_2 2.0) (- 1.0 (* -0.5 t_1))) 3.0))
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4.6e-6) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_0)))) / ((1.0 + fma(0.5, (cos(x) * t_1), (0.5 * t_2))) * 3.0);
	} else if (x <= 7.7e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / (fma(cos(y), (t_2 / 2.0), (1.0 - (-0.5 * t_1))) * 3.0);
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4.6e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_0)))) / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * t_2))) * 3.0));
	elseif (x <= 7.7e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(fma(cos(y), Float64(t_2 / 2.0), Float64(1.0 - Float64(-0.5 * t_1))) * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-6], 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.7e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(1.0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $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$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-6

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 61.2%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 60.1

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

   9. Applied rewrites 60.1%

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

   if -4.6e-6 < x < 7.7000000000000004e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if 7.7000000000000004e-7 < x

   1. Initial program 99.0%

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

   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 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   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. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 34: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot t\_2\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_2, 0.5 \cdot t\_1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -4.6e-6)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0))))
      (* (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 t_2))) 3.0))
     (if (<= x 7.7e-7)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ y y)))) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* (+ 1.0 (fma 0.5 (* (cos y) t_2) (* 0.5 t_1))) 3.0))
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4.6e-6) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_0)))) / ((1.0 + fma(0.5, (cos(x) * t_1), (0.5 * t_2))) * 3.0);
	} else if (x <= 7.7e-7) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((y + y)))) * (sqrt(2.0) * (1.0 - cos(y)))))) / ((1.0 + fma(0.5, (cos(y) * t_2), (0.5 * t_1))) * 3.0);
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4.6e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_0)))) / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * t_2))) * 3.0));
	elseif (x <= 7.7e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(Float64(1.0 + fma(0.5, Float64(cos(y) * t_2), Float64(0.5 * t_1))) * 3.0));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-6], 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.7e-7], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $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$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-6

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 61.2%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 60.1

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

   9. Applied rewrites 60.1%

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

   if -4.6e-6 < x < 7.7000000000000004e-7

   1. Initial program 99.6%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if 7.7000000000000004e-7 < x

   1. Initial program 99.0%

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

   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 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   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. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 35: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_1, 0.5 \cdot t\_2\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -4.6e-6)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0))))
      (* (+ 1.0 (fma 0.5 (* (cos x) t_1) (* 0.5 t_2))) 3.0))
     (if (<= x 7.7e-7)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_1) 1.0))
        0.3333333333333333)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4.6e-6) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_0)))) / ((1.0 + fma(0.5, (cos(x) * t_1), (0.5 * t_2))) * 3.0);
	} else if (x <= 7.7e-7) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4.6e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_0)))) / Float64(Float64(1.0 + fma(0.5, Float64(cos(x) * t_1), Float64(0.5 * t_2))) * 3.0));
	elseif (x <= 7.7e-7)
		tmp = Float64(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) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-6], 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.7e-7], N[(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[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-6

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 61.2%

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

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 60.1

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

   9. Applied rewrites 60.1%

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

   if -4.6e-6 < x < 7.7000000000000004e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   if 7.7000000000000004e-7 < x

   1. Initial program 99.0%

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

   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 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   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. lower--.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-cos.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 36: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -4.6e-6)
     (*
      (/
       (fma (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x))))) t_0) (sqrt 2.0) 2.0)
       (fma 0.5 (fma t_1 (cos x) t_2) 1.0))
      0.3333333333333333)
     (if (<= x 7.7e-7)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_1) 1.0))
        0.3333333333333333)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* t_0 (sqrt 2.0))
          2.0)
         (fma 0.5 (- (fma t_1 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -4.6e-6) {
		tmp = (fma(((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * t_0), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333;
	} else if (x <= 7.7e-7) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_0 * sqrt(2.0)), 2.0) / fma(0.5, (fma(t_1, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -4.6e-6)
		tmp = Float64(Float64(fma(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * t_0), sqrt(2.0), 2.0) / fma(0.5, fma(t_1, cos(x), t_2), 1.0)) * 0.3333333333333333);
	elseif (x <= 7.7e-7)
		tmp = Float64(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) / fma(0.5, fma(t_2, cos(y), t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_0 * sqrt(2.0)), 2.0) / fma(0.5, Float64(fma(t_1, cos(x), 3.0) - sqrt(5.0)), 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-6], N[(N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 7.7e-7], N[(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[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e-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. 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 60.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.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. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2} + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 60.1%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -4.6e-6 < x < 7.7000000000000004e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 7.7000000000000004e-7 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   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. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 59.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 59.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 37: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
    (* (- (cos x) 1.0) (sqrt 2.0))
    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 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 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(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(fma(Float64(sqrt(5.0) - 1.0), cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := 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[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
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. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \left(\left(\sqrt{5} - 1\right) \cdot \cos x + 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot \frac{1}{3} \]

   11. lift-cos.f64 60.5

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

6. Applied rewrites 60.5%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 38: 60.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x))))) (- (cos x) 1.0))
    (sqrt 2.0)
    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 * (0.5 - (0.5 * cos((x + x))))) * (cos(x) - 1.0)), sqrt(2.0), 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(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(cos(x) - 1.0)), sqrt(2.0), 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[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $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[(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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   5. lift-*.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. lift-cos.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   8. lift--.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   9. lift-cos.f64 N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   11. associate-*r* N/A

       \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2} + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

6. Applied rewrites 60.5%

   \[\leadsto \frac{\mathsf{fma}\left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 39: 45.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (fma
    (cos y)
    (/ (- 3.0 (sqrt 5.0)) 2.0)
    (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(cos(y), ((3.0 - sqrt(5.0)) / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}} \]

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
5. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   3. metadata-eval N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin y}^{2}} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   5. unpow2 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin y \cdot \sin y\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. sqr-sin-a-rev N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

6. Applied rewrites 62.7%

   \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

7. Taylor expanded in y around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
8. Step-by-step derivation

9. Applied rewrites 45.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
10. Add Preprocessing

Alternative 40: 43.4% 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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

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 43.4%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 41: 40.9% 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 60.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.9%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(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))))))