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

Percentage Accurate: 99.3% → 99.3%

Time: 12.9s

Alternatives: 33

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos x) (cos y))
   (* (- (sin y) (/ (sin x) 16.0)) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
   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)), ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 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(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), 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[(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[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. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-sin.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

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

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

    2. lower--.f64 N/A

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

    3. metadata-eval N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lift-cos.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. lift--.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (*
    (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* (sin y) 0.0625)))
    (- (cos x) (cos y)))
   (sqrt 2.0)
   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((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625))) * (cos(x) - cos(y))), sqrt(2.0), 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(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(cos(x) - cos(y))), sqrt(2.0), 2.0) / Float64(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[(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[(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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-cos.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-sin.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around inf

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

    1. lower-+.f64 N/A

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

    2. distribute-lft-out N/A

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

    3. lower-*.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-cos.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift--.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-cos.f64 N/A

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

    10. lift-sqrt.f64 N/A

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

    11. lift--.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

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 6: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5\right) \cdot x, x, 1\right)\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_3\right) \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), t\_2 - \cos y, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(t\_2, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

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 (fma (* (- (* (* x x) 0.041666666666666664) 0.5) x) x 1.0))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_3) (- (cos x) (cos y))))))
   (if (<= x -0.162)
     (/ t_4 (fma (fma (cos x) t_1 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.17)
       (/
        (fma
         (* t_3 (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
         (- t_2 (cos y))
         2.0)
        (* (fma (cos y) t_0 (fma t_2 t_1 1.0)) 3.0))
       (/ t_4 (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* 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 = fma(((((x * x) * 0.041666666666666664) - 0.5) * x), x, 1.0);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = 2.0 + (((sqrt(2.0) * sin(x)) * t_3) * (cos(x) - cos(y)));
	double tmp;
	if (x <= -0.162) {
		tmp = t_4 / fma(fma(cos(x), t_1, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.17) {
		tmp = fma((t_3 * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), (t_2 - cos(y)), 2.0) / (fma(cos(y), t_0, fma(t_2, t_1, 1.0)) * 3.0);
	} else {
		tmp = t_4 / (3.0 * ((1.0 + (t_1 * cos(x))) + (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 = fma(Float64(Float64(Float64(Float64(x * x) * 0.041666666666666664) - 0.5) * x), x, 1.0)
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_3) * Float64(cos(x) - cos(y))))
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(t_4 / fma(fma(cos(x), t_1, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.17)
		tmp = Float64(fma(Float64(t_3 * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))), Float64(t_2 - cos(y)), 2.0) / Float64(fma(cos(y), t_0, fma(t_2, t_1, 1.0)) * 3.0));
	else
		tmp = Float64(t_4 / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + 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[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(t$95$4 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(N[(t$95$3 * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[Cos[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(t$95$2 * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$4 / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.5%

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

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 7: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := 2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ t_4 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\\ t_5 := t\_0 \cdot \cos y\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_3, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_4 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_4}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot t\_1\right) + t\_5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2
         (+
          2.0
          (*
           (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
           (- (cos x) (cos y)))))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_4
         (*
          (fma
           (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
           (* x x)
           1.0)
          x))
        (t_5 (* t_0 (cos y))))
   (if (<= x -0.162)
     (/ t_2 (fma (fma (cos x) t_3 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_4 (/ (sin y) 16.0))) (- (sin y) (/ t_4 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 t_1)) t_5)))
       (/ t_2 (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) t_5)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = 2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)));
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double t_4 = fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x;
	double t_5 = t_0 * cos(y);
	double tmp;
	if (x <= -0.162) {
		tmp = t_2 / fma(fma(cos(x), t_3, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_4 - (sin(y) / 16.0))) * (sin(y) - (t_4 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_3 * t_1)) + t_5));
	} else {
		tmp = t_2 / (3.0 * ((1.0 + (t_3 * cos(x))) + t_5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))))
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_4 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)
	t_5 = Float64(t_0 * cos(y))
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(t_2 / fma(fma(cos(x), t_3, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_4 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_4 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * t_1)) + t_5)));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + t_5)));
	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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(t$95$2 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$4 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 99.5

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 8: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ t_2 := \cos x - \cos y\\ t_3 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\\ t_4 := \sin y - \frac{\sin x}{16}\\ t_5 := \frac{\sqrt{5} - 1}{2}\\ t_6 := t\_1 \cdot \cos y\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_4 \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_5, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_3 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_3}{16}\right)\right) \cdot \left(t\_0 - \cos y\right)}{3 \cdot \left(\left(1 + t\_5 \cdot t\_0\right) + t\_6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t\_4\right) \cdot t\_2}{3 \cdot \left(\left(1 + t\_5 \cdot \cos x\right) + t\_6\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (*
          (fma
           (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
           (* x x)
           1.0)
          x))
        (t_4 (- (sin y) (/ (sin x) 16.0)))
        (t_5 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_6 (* t_1 (cos y))))
   (if (<= x -0.162)
     (/
      (fma t_2 (* t_4 (* (sin x) (sqrt 2.0))) 2.0)
      (* (fma (cos y) t_1 (fma (cos x) t_5 1.0)) 3.0))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_3 (/ (sin y) 16.0))) (- (sin y) (/ t_3 16.0)))
          (- t_0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_5 t_0)) t_6)))
       (/
        (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) t_4) t_2))
        (* 3.0 (+ (+ 1.0 (* t_5 (cos x))) t_6)))))))

C

double code(double x, double y) {
	double t_0 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_1 = (3.0 - sqrt(5.0)) / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x;
	double t_4 = sin(y) - (sin(x) / 16.0);
	double t_5 = (sqrt(5.0) - 1.0) / 2.0;
	double t_6 = t_1 * cos(y);
	double tmp;
	if (x <= -0.162) {
		tmp = fma(t_2, (t_4 * (sin(x) * sqrt(2.0))), 2.0) / (fma(cos(y), t_1, fma(cos(x), t_5, 1.0)) * 3.0);
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_3 - (sin(y) / 16.0))) * (sin(y) - (t_3 / 16.0))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + (t_5 * t_0)) + t_6));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * t_4) * t_2)) / (3.0 * ((1.0 + (t_5 * cos(x))) + t_6));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)
	t_4 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_5 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_6 = Float64(t_1 * cos(y))
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(fma(t_2, Float64(t_4 * Float64(sin(x) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), t_1, fma(cos(x), t_5, 1.0)) * 3.0));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_3 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_3 / 16.0))) * Float64(t_0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_5 * t_0)) + t_6)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * t_4) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_5 * cos(x))) + t_6)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(N[(t$95$2 * N[(t$95$4 * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$5 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$5 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      1. lift-sin.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 99.5

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 9: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \sqrt{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \frac{3 - \sqrt{5}}{2}\\ t_3 := \cos x - \cos y\\ t_4 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\\ t_5 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, \left(\sin y - \frac{\sin x}{16}\right) \cdot t\_0, 2\right)}{\mathsf{fma}\left(\cos y, t\_2, \mathsf{fma}\left(\cos x, t\_5, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_4 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_4}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_5 \cdot t\_1\right) + t\_2 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, t\_0 \cdot \left(\sin y - \sin x \cdot 0.0625\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_5 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (sqrt 2.0)))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_3 (- (cos x) (cos y)))
        (t_4
         (*
          (fma
           (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
           (* x x)
           1.0)
          x))
        (t_5 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (<= x -0.162)
     (/
      (fma t_3 (* (- (sin y) (/ (sin x) 16.0)) t_0) 2.0)
      (* (fma (cos y) t_2 (fma (cos x) t_5 1.0)) 3.0))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_4 (/ (sin y) 16.0))) (- (sin y) (/ t_4 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_5 t_1)) (* t_2 (cos y)))))
       (/
        (fma t_3 (* t_0 (- (sin y) (* (sin x) 0.0625))) 2.0)
        (* (fma (cos y) t_2 (+ (* (cos x) t_5) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * sqrt(2.0);
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = (3.0 - sqrt(5.0)) / 2.0;
	double t_3 = cos(x) - cos(y);
	double t_4 = fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x;
	double t_5 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if (x <= -0.162) {
		tmp = fma(t_3, ((sin(y) - (sin(x) / 16.0)) * t_0), 2.0) / (fma(cos(y), t_2, fma(cos(x), t_5, 1.0)) * 3.0);
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_4 - (sin(y) / 16.0))) * (sin(y) - (t_4 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_5 * t_1)) + (t_2 * cos(y))));
	} else {
		tmp = fma(t_3, (t_0 * (sin(y) - (sin(x) * 0.0625))), 2.0) / (fma(cos(y), t_2, ((cos(x) * t_5) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * sqrt(2.0))
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)
	t_5 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(fma(t_3, Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0), 2.0) / Float64(fma(cos(y), t_2, fma(cos(x), t_5, 1.0)) * 3.0));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_4 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_4 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_5 * t_1)) + Float64(t_2 * cos(y)))));
	else
		tmp = Float64(fma(t_3, Float64(t_0 * Float64(sin(y) - Float64(sin(x) * 0.0625))), 2.0) / Float64(fma(cos(y), t_2, Float64(Float64(cos(x) * t_5) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(N[(t$95$3 * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$5 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$4 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      1. lift-sin.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 99.5

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 98.9

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in y around 0

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

       1. lift-sin.f64 62.6

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

   11. Applied rewrites 62.6%

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

Alternative 10: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ t_2 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ t_4 := \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_2 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_2}{16}\right)\right) \cdot \left(t\_0 - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot t\_0\right) + t\_1 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_2
         (*
          (fma
           (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
           (* x x)
           1.0)
          x))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_4
         (/
          (fma
           (- (cos x) (cos y))
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           2.0)
          (* (fma (cos y) t_1 (fma (cos x) t_3 1.0)) 3.0))))
   (if (<= x -0.162)
     t_4
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_2 (/ (sin y) 16.0))) (- (sin y) (/ t_2 16.0)))
          (- t_0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 t_0)) (* t_1 (cos y)))))
       t_4))))

C

double code(double x, double y) {
	double t_0 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_1 = (3.0 - sqrt(5.0)) / 2.0;
	double t_2 = fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x;
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double t_4 = fma((cos(x) - cos(y)), ((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))), 2.0) / (fma(cos(y), t_1, fma(cos(x), t_3, 1.0)) * 3.0);
	double tmp;
	if (x <= -0.162) {
		tmp = t_4;
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_2 - (sin(y) / 16.0))) * (sin(y) - (t_2 / 16.0))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + (t_3 * t_0)) + (t_1 * cos(y))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_2 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_4 = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))), 2.0) / Float64(fma(cos(y), t_1, fma(cos(x), t_3, 1.0)) * 3.0))
	tmp = 0.0
	if (x <= -0.162)
		tmp = t_4;
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_2 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_2 / 16.0))) * Float64(t_0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * t_0)) + Float64(t_1 * cos(y)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = 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[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.162], t$95$4, If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$2 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.162000000000000005 or 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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

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

      1. lift-sin.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 99.5

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ t_4 := \frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.14:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y \leq 0.014:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_2}{16}\right)\right) \cdot \left(t\_2 - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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 (* y y) -0.16666666666666666 1.0) y))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_4
         (/
          (fma t_1 (* (sin y) (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))) 2.0)
          (* (fma (cos y) t_0 (fma (cos x) t_3 1.0)) 3.0))))
   (if (<= y -0.14)
     t_4
     (if (<= y 0.014)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_2 16.0))) (- t_2 (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) (* t_0 (cos y)))))
       t_4))))

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((y * y), -0.16666666666666666, 1.0) * y;
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double t_4 = fma(t_1, (sin(y) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_3, 1.0)) * 3.0);
	double tmp;
	if (y <= -0.14) {
		tmp = t_4;
	} else if (y <= 0.014) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_2 / 16.0))) * (t_2 - (sin(x) / 16.0))) * t_1)) / (3.0 * ((1.0 + (t_3 * cos(x))) + (t_0 * cos(y))));
	} else {
		tmp = t_4;
	}
	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 = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y)
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_4 = 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, fma(cos(x), t_3, 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.14)
		tmp = t_4;
	elseif (y <= 0.014)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_2 / 16.0))) * Float64(t_2 - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + Float64(t_0 * cos(y)))));
	else
		tmp = t_4;
	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[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = 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 + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.14], t$95$4, If[LessEqual[y, 0.014], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(t$95$2 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.14000000000000001 or 0.0140000000000000003 < y

   1. Initial program 99.0%

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

   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 63.1

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

   6. Applied rewrites 63.1%

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

   if -0.14000000000000001 < y < 0.0140000000000000003

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{16}\right)\right) \cdot \left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y} - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{if}\;x \leq -0.222:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_3 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_3}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3
         (*
          (fma
           (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
           (* x x)
           1.0)
          x)))
   (if (<= x -0.222)
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) 1.0)))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_3 (/ (sin y) 16.0))) (- (sin y) (/ t_3 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* (cos (+ x x)) 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x;
	double tmp;
	if (x <= -0.222) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_3 - (sin(y) / 16.0))) * (sin(y) - (t_3 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (cos((x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)
	tmp = 0.0
	if (x <= -0.222)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_3 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_3 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.222], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.222000000000000003

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 59.9

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

   6. Applied rewrites 59.9%

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

   7. Taylor expanded in y around 0

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

      1. lift-sin.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if -0.222000000000000003 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 99.5

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 59.0

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

   8. Applied rewrites 59.0%

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

Alternative 13: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{t\_0}{2}\\ t_4 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_5 := \frac{t\_2}{2}\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_4 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_4}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_5 \cdot t\_1\right) + t\_3 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_3, \cos x \cdot t\_5 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (/ t_0 2.0))
        (t_4 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_5 (/ t_2 2.0)))
   (if (<= x -0.162)
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) 1.0)))
      (fma (fma (* 0.5 (cos x)) t_2 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_4 (/ (sin y) 16.0))) (- (sin y) (/ t_4 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_5 t_1)) (* t_3 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* (cos (+ x x)) 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_3 (+ (* (cos x) t_5) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = t_0 / 2.0;
	double t_4 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_5 = t_2 / 2.0;
	double tmp;
	if (x <= -0.162) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / fma(fma((0.5 * cos(x)), t_2, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_4 - (sin(y) / 16.0))) * (sin(y) - (t_4 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_5 * t_1)) + (t_3 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (cos((x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_3, ((cos(x) * t_5) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(t_0 / 2.0)
	t_4 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_5 = Float64(t_2 / 2.0)
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / fma(fma(Float64(0.5 * cos(x)), t_2, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_4 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_4 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_5 * t_1)) + Float64(t_3 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_3, Float64(Float64(cos(x) * t_5) + 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$4 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$5 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$3 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 59.9

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

   6. Applied rewrites 59.9%

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

   7. Taylor expanded in x around inf

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

      1. distribute-rgt-out N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 59.9%

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

   10. Taylor expanded in y around 0

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

       1. lift-sin.f64 59.4

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

   12. Applied rewrites 59.4%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 59.0

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

   8. Applied rewrites 59.0%

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

Alternative 14: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_3 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_3}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (* (fma (* x x) -0.16666666666666666 1.0) x)))
   (if (<= x -0.162)
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) 1.0)))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_3 (/ (sin y) 16.0))) (- (sin y) (/ t_3 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* (cos (+ x x)) 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double tmp;
	if (x <= -0.162) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_3 - (sin(y) / 16.0))) * (sin(y) - (t_3 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (cos((x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_3 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_3 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -0.162], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 59.9

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

   6. Applied rewrites 59.9%

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

   7. Taylor expanded in y around 0

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

      1. lift-sin.f64 59.4

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

   9. Applied rewrites 59.4%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 59.0

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

   8. Applied rewrites 59.0%

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

Alternative 15: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_4 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -0.162:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_4\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.17:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_3 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_3}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - t\_4 \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_4 (cos (+ x x))))
   (if (<= x -0.162)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_4)) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.17)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_3 (/ (sin y) 16.0))) (- (sin y) (/ t_3 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* t_4 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_4 = cos((x + x));
	double tmp;
	if (x <= -0.162) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * t_4)) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.17) {
		tmp = (2.0 + (((sqrt(2.0) * (t_3 - (sin(y) / 16.0))) * (sin(y) - (t_3 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_4 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -0.162)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_4)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.17)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_3 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_3 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.162], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.17], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.162000000000000005

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 59.9

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

   6. Applied rewrites 59.9%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 26.1

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

   9. Applied rewrites 26.1%

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

   10. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

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

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

       7. lower-*.f64 N/A

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

   12. Applied rewrites 59.1%

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

   if -0.162000000000000005 < x < 0.170000000000000012

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.170000000000000012 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-sqrt.f64 59.0

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

   8. Applied rewrites 59.0%

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

Alternative 16: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ t_4 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_4\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_3 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_3}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - t\_4 \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (* x x) -0.5 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (* (fma (* x x) -0.16666666666666666 1.0) x))
        (t_4 (cos (+ x x))))
   (if (<= x -0.024)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_4)) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.026)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_3 (/ (sin y) 16.0))) (- (sin y) (/ t_3 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* t_4 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma((x * x), -0.5, 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma((x * x), -0.16666666666666666, 1.0) * x;
	double t_4 = cos((x + x));
	double tmp;
	if (x <= -0.024) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * t_4)) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.026) {
		tmp = (2.0 + (((sqrt(2.0) * (t_3 - (sin(y) / 16.0))) * (sin(y) - (t_3 / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(x * x), -0.5, 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
	t_4 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -0.024)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_4)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_3 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_3 / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.024], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$3 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   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

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

      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 99.0%

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

   4. Taylor expanded in y around 0

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 59.8

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

   6. Applied rewrites 59.8%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 26.2

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

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

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

       7. lower-*.f64 N/A

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

   12. Applied rewrites 59.1%

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

   if -0.024 < x < 0.0259999999999999988

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0259999999999999988 < x

   1. Initial program 98.9%

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

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      12. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      14. lift-sqrt.f64 58.9

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

   8. Applied rewrites 58.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -0.043:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_3\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (cos (+ x x))))
   (if (<= x -0.043)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_3)) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.045)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* t_3 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = cos((x + x));
	double tmp;
	if (x <= -0.043) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * t_3)) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.045) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (t_3 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -0.043)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_3)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.045)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.043], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.045], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.042999999999999997

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

      2. lift--.f64 59.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   6. Applied rewrites 59.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 26.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   9. Applied rewrites 26.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin x}^{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   12. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   if -0.042999999999999997 < x < 0.044999999999999998

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.044999999999999998 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1}\right) \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + 1\right) \cdot 3} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) \cdot 3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + 1\right) \cdot 3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x + 1\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}\right) \cdot 3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x + 1\right) \cdot 3} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + 1\right) \cdot 3} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + 1\right) \cdot 3} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + 1\right) \cdot 3} \]

      13. lift-cos.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x} \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1}\right) \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      12. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      14. lift-sqrt.f64 59.0

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

   8. Applied rewrites 59.0%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_3\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2 + 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (* x x) -0.5 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (cos (+ x x))))
   (if (<= x -0.023)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_3)) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma (fma (cos x) t_2 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.026)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* t_3 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 (+ (* (cos x) t_2) 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma((x * x), -0.5, 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = cos((x + x));
	double tmp;
	if (x <= -0.023) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * t_3)) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.026) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (t_3 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, ((cos(x) * t_2) + 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(x * x), -0.5, 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -0.023)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_3)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, Float64(Float64(cos(x) * t_2) + 1.0)) * 3.0));
	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[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.023], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.023

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

      2. lift--.f64 59.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   6. Applied rewrites 59.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 26.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   9. Applied rewrites 26.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin x}^{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   12. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   if -0.023 < x < 0.0259999999999999988

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0259999999999999988 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1}\right) \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + 1\right) \cdot 3} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) \cdot 3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + 1\right) \cdot 3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x + 1\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}\right) \cdot 3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x + 1\right) \cdot 3} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + 1\right) \cdot 3} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + 1\right) \cdot 3} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + 1\right) \cdot 3} \]

      13. lift-cos.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x} \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2} + 1}\right) \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      12. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

      14. lift-sqrt.f64 58.9

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]

   8. Applied rewrites 58.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \cos x \cdot \frac{\sqrt{5} - 1}{2} + 1\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \mathsf{fma}\left(\cos x, t\_2, 1\right)\\ t_4 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot t\_4\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(t\_3, 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - t\_4 \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, t\_3\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (* x x) -0.5 1.0))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (fma (cos x) t_2 1.0))
        (t_4 (cos (+ x x))))
   (if (<= x -0.023)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 t_4)) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma t_3 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.026)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) (* t_0 (cos y)))))
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (- 0.5 (* t_4 0.5)) (sqrt 2.0)) -0.0625)
         2.0)
        (* (fma (cos y) t_0 t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma((x * x), -0.5, 1.0);
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = fma(cos(x), t_2, 1.0);
	double t_4 = cos((x + x));
	double tmp;
	if (x <= -0.023) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * t_4)) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(t_3, 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.026) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + (t_0 * cos(y))));
	} else {
		tmp = fma((cos(x) - cos(y)), (((0.5 - (t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / (fma(cos(y), t_0, t_3) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(x * x), -0.5, 1.0)
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = fma(cos(x), t_2, 1.0)
	t_4 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= -0.023)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * t_4)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(t_3, 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + Float64(t_0 * cos(y)))));
	else
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(Float64(0.5 - Float64(t_4 * 0.5)) * sqrt(2.0)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, t_3) * 3.0));
	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[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.023], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 - N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.023

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

      2. lift--.f64 59.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   6. Applied rewrites 59.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 26.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   9. Applied rewrites 26.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin x}^{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   12. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   if -0.023 < x < 0.0259999999999999988

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0259999999999999988 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{-1}{16}}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      10. count-2-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(\frac{1}{2} - \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \sqrt{2}\right) \cdot \frac{-1}{16}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      12. lift-sqrt.f64 58.9

          \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 58.9%

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \color{blue}{\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot \sqrt{2}\right) \cdot -0.0625}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ t_2 := \cos x - 1\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ t_4 := t\_0 \cdot \cos y\\ \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot t\_2\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_3, 1\right), 3, \left(\cos y \cdot t\_0\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_3 \cdot t\_1\right) + t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_2 \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (* x x) -0.5 1.0))
        (t_2 (- (cos x) 1.0))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_4 (* t_0 (cos y))))
   (if (<= x -0.023)
     (/
      (- 2.0 (* 0.0625 (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_2))))
      (fma (fma (cos x) t_3 1.0) 3.0 (* (* (cos y) t_0) 3.0)))
     (if (<= x 0.026)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_3 t_1)) t_4)))
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         (* t_2 (sqrt 2.0))
         2.0)
        (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) t_4)))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma((x * x), -0.5, 1.0);
	double t_2 = cos(x) - 1.0;
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double t_4 = t_0 * cos(y);
	double tmp;
	if (x <= -0.023) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * t_2)))) / fma(fma(cos(x), t_3, 1.0), 3.0, ((cos(y) * t_0) * 3.0));
	} else if (x <= 0.026) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_3 * t_1)) + t_4));
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), (t_2 * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + (t_3 * cos(x))) + t_4));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(x * x), -0.5, 1.0)
	t_2 = Float64(cos(x) - 1.0)
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_4 = Float64(t_0 * cos(y))
	tmp = 0.0
	if (x <= -0.023)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * t_2)))) / fma(fma(cos(x), t_3, 1.0), 3.0, Float64(Float64(cos(y) * t_0) * 3.0)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * t_1)) + t_4)));
	else
		tmp = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(t_2 * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + t_4)));
	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[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.023], 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$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.026], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.023

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

      2. lift--.f64 59.8

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   6. Applied rewrites 59.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 26.2

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   9. Applied rewrites 26.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin x}^{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       6. sqr-sin-a-rev N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   12. Applied rewrites 59.1%

       \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   if -0.023 < x < 0.0259999999999999988

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{x} - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{x}}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.0259999999999999988 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2} \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 58.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos y\\ t_1 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{t\_2}{2}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{\mathsf{fma}\left(t\_1, 3, \left(\cos y \cdot t\_3\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00145:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), t\_0 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, t\_3, t\_1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos y)))
        (t_1 (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (/ t_2 2.0)))
   (if (<= y -4.4e-6)
     (/
      (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) t_0))))
      (fma t_1 3.0 (* (* (cos y) t_3) 3.0)))
     (if (<= y 0.00145)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma (expm1 (* (log 5.0) 0.5)) (cos x) t_2) 1.0))
        0.3333333333333333)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* t_0 (sqrt 2.0))
         2.0)
        (* (fma (cos y) t_3 t_1) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - cos(y);
	double t_1 = fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = t_2 / 2.0;
	double tmp;
	if (y <= -4.4e-6) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * t_0)))) / fma(t_1, 3.0, ((cos(y) * t_3) * 3.0));
	} else if (y <= 0.00145) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1((log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), (t_0 * sqrt(2.0)), 2.0) / (fma(cos(y), t_3, t_1) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - cos(y))
	t_1 = fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(t_2 / 2.0)
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * t_0)))) / fma(t_1, 3.0, Float64(Float64(cos(y) * t_3) * 3.0)));
	elseif (y <= 0.00145)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1(Float64(log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333);
	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(fma(cos(y), t_3, t_1) * 3.0));
	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[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / 2.0), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00145], 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[(Exp[N[(N[Log[5.0], $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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[(N[(N[Cos[y], $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000002e-6

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

      2. lift--.f64 27.1

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   6. Applied rewrites 27.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. lift-sin.f64 26.5

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   9. Applied rewrites 26.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left(\color{blue}{{\sin y}^{2}} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       7. lift-sin.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       9. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

       11. lift-cos.f64 59.8

           \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   12. Applied rewrites 59.8%

       \[\leadsto \frac{\color{blue}{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)} \]

   if -4.4000000000000002e-6 < y < 0.00145

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left({5}^{\frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(e^{\log 5 \cdot \frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-log.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if 0.00145 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 59.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{t\_0}{2}\\ t_2 := \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_3 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_2}{3 \cdot \left(\left(1 + t\_3 \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \mathbf{elif}\;y \leq 0.00145:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(\cos y, t\_1, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (/ t_0 2.0))
        (t_2
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_3 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (<= y -4.4e-6)
     (/ t_2 (* 3.0 (+ (+ 1.0 (* t_3 (cos x))) (* t_1 (cos y)))))
     (if (<= y 0.00145)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma (expm1 (* (log 5.0) 0.5)) (cos x) t_0) 1.0))
        0.3333333333333333)
       (/ t_2 (* (fma (cos y) t_1 (fma (cos x) t_3 1.0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = t_0 / 2.0;
	double t_2 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_3 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if (y <= -4.4e-6) {
		tmp = t_2 / (3.0 * ((1.0 + (t_3 * cos(x))) + (t_1 * cos(y))));
	} else if (y <= 0.00145) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1((log(5.0) * 0.5)), cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2 / (fma(cos(y), t_1, fma(cos(x), t_3, 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(t_0 / 2.0)
	t_2 = 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)
	t_3 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(t_3 * cos(x))) + Float64(t_1 * cos(y)))));
	elseif (y <= 0.00145)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1(Float64(log(5.0) * 0.5)), cos(x), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_2 / Float64(fma(cos(y), t_1, fma(cos(x), t_3, 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[y, -4.4e-6], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00145], 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[(Exp[N[(N[Log[5.0], $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$2 / N[(N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4000000000000002e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 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 59.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -4.4000000000000002e-6 < y < 0.00145

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left({5}^{\frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(e^{\log 5 \cdot \frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-log.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if 0.00145 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 59.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{t\_0}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00145:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           (fma
            (cos y)
            (/ t_0 2.0)
            (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0))
           3.0))))
   (if (<= y -4.4e-6)
     t_1
     (if (<= y 0.00145)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma (expm1 (* (log 5.0) 0.5)) (cos x) t_0) 1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(cos(y), (t_0 / 2.0), fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0);
	double tmp;
	if (y <= -4.4e-6) {
		tmp = t_1;
	} else if (y <= 0.00145) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1((log(5.0) * 0.5)), cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = 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(fma(cos(y), Float64(t_0 / 2.0), fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -4.4e-6)
		tmp = t_1;
	elseif (y <= 0.00145)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(expm1(Float64(log(5.0) * 0.5)), cos(x), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-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[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 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]}, If[LessEqual[y, -4.4e-6], t$95$1, If[LessEqual[y, 0.00145], 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[(Exp[N[(N[Log[5.0], $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000002e-6 or 0.00145 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   if -4.4000000000000002e-6 < y < 0.00145

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left({5}^{\frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(e^{\log 5 \cdot \frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-log.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{\mathsf{fma}\left(\cos y, t\_0, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + t\_1 \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + t\_0 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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
         (/
          (fma
           (* (- (cos x) 1.0) (sqrt 2.0))
           (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
           2.0)
          (* (fma (cos y) t_0 (fma (cos x) t_1 1.0)) 3.0))))
   (if (<= x -0.0082)
     t_2
     (if (<= x 0.0011)
       (/
        (+
         2.0
         (*
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))))
        (* 3.0 (+ (+ 1.0 (* t_1 (fma (* x x) -0.5 1.0))) (* t_0 (cos y)))))
       t_2))))

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 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) / (fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0);
	double tmp;
	if (x <= -0.0082) {
		tmp = t_2;
	} else if (x <= 0.0011) {
		tmp = (2.0 + ((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))) * ((1.0 - cos(y)) * sqrt(2.0)))) / (3.0 * ((1.0 + (t_1 * fma((x * x), -0.5, 1.0))) + (t_0 * cos(y))));
	} else {
		tmp = t_2;
	}
	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(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) / Float64(fma(cos(y), t_0, fma(cos(x), t_1, 1.0)) * 3.0))
	tmp = 0.0
	if (x <= -0.0082)
		tmp = t_2;
	elseif (x <= 0.0011)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * fma(Float64(x * x), -0.5, 1.0))) + Float64(t_0 * cos(y)))));
	else
		tmp = t_2;
	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[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $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]}, If[LessEqual[x, -0.0082], t$95$2, If[LessEqual[x, 0.0011], N[(N[(2.0 + 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00820000000000000069 or 0.00110000000000000007 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) + 2}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}, 2\right)}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   6. Applied rewrites 59.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}}{\mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{2}, \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right)\right) \cdot 3} \]

   if -0.00820000000000000069 < x < 0.00110000000000000007

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sqrt.f64 98.7

          \[\leadsto \frac{2 + \left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 98.7%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 0.0021:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0082)
     (*
      (/ t_0 (+ (fma (* 0.5 (cos x)) t_1 1.0) (* 0.5 t_2)))
      0.3333333333333333)
     (if (<= x 0.0021)
       (/
        (+
         2.0
         (*
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_1 2.0) (fma (* x x) -0.5 1.0)))
          (* (/ t_2 2.0) (cos y)))))
       (*
        (/ t_0 (fma 0.5 (fma (expm1 (* (log 5.0) 0.5)) (cos x) t_2) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0082) {
		tmp = (t_0 / (fma((0.5 * cos(x)), t_1, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	} else if (x <= 0.0021) {
		tmp = (2.0 + ((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))) * ((1.0 - cos(y)) * sqrt(2.0)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * fma((x * x), -0.5, 1.0))) + ((t_2 / 2.0) * cos(y))));
	} else {
		tmp = (t_0 / fma(0.5, fma(expm1((log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = 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)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(t_0 / Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(0.5 * t_2))) * 0.3333333333333333);
	elseif (x <= 0.0021)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * fma(Float64(x * x), -0.5, 1.0))) + Float64(Float64(t_2 / 2.0) * cos(y)))));
	else
		tmp = Float64(Float64(t_0 / fma(0.5, fma(expm1(Float64(log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, 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, -0.0082], N[(N[(t$95$0 / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 0.0021], N[(N[(2.0 + 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(0.5 * N[(N[(Exp[N[(N[Log[5.0], $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   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 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 57.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333 \]

   if -0.00820000000000000069 < x < 0.00209999999999999987

   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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left({x}^{2} \cdot \frac{-1}{2} + 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left(\frac{-1}{2} \cdot {x}^{2} + \color{blue}{1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \left({x}^{2} \cdot \frac{-1}{2} + 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{2}}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{\color{blue}{2}}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-sqrt.f64 98.7

          \[\leadsto \frac{2 + \left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 98.7%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right)\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \mathsf{fma}\left(x \cdot x, -0.5, 1\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.00209999999999999987 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift--.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-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left({5}^{\frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(e^{\log 5 \cdot \frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-log.f64 58.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0082)
     (*
      (/ t_0 (+ (fma (* 0.5 (cos x)) t_1 1.0) (* 0.5 t_2)))
      0.3333333333333333)
     (if (<= x 1.05e-5)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (* 1.5 (cos y)) t_2 (* (fma 0.5 t_1 1.0) 3.0)))
       (*
        (/ t_0 (fma 0.5 (fma (expm1 (* (log 5.0) 0.5)) (cos x) t_2) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0082) {
		tmp = (t_0 / (fma((0.5 * cos(x)), t_1, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	} else if (x <= 1.05e-5) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_2, (fma(0.5, t_1, 1.0) * 3.0));
	} else {
		tmp = (t_0 / fma(0.5, fma(expm1((log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = 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)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(t_0 / Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(0.5 * t_2))) * 0.3333333333333333);
	elseif (x <= 1.05e-5)
		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) / fma(Float64(1.5 * cos(y)), t_2, Float64(fma(0.5, t_1, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(t_0 / fma(0.5, fma(expm1(Float64(log(5.0) * 0.5)), cos(x), t_2), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, 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, -0.0082], N[(N[(t$95$0 / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.05e-5], 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[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(0.5 * N[(N[(Exp[N[(N[Log[5.0], $MachinePrecision] * 0.5), $MachinePrecision]] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   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 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 57.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333 \]

   if -0.00820000000000000069 < x < 1.04999999999999994e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}} \]

   if 1.04999999999999994e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift--.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-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left({5}^{\frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(e^{\log 5 \cdot \frac{1}{2}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot \frac{1}{2}\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-log.f64 58.0

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\mathsf{expm1}\left(\log 5 \cdot 0.5\right), \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 77.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_0, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right) + 0.5 \cdot t\_2} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, t\_2\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0082)
     (*
      (/
       (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0 2.0)
       (+ (fma (* 0.5 (cos x)) t_1 1.0) (* 0.5 t_2)))
      0.3333333333333333)
     (if (<= x 1.05e-5)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (* 1.5 (cos y)) t_2 (* (fma 0.5 t_1 1.0) 3.0)))
       (/
        (*
         (fma t_0 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) 2.0)
         0.3333333333333333)
        (fma (fma t_1 (cos x) t_2) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0082) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_0, 2.0) / (fma((0.5 * cos(x)), t_1, 1.0) + (0.5 * t_2))) * 0.3333333333333333;
	} else if (x <= 1.05e-5) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_2, (fma(0.5, t_1, 1.0) * 3.0));
	} else {
		tmp = (fma(t_0, ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_1, cos(x), t_2), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_0, 2.0) / Float64(fma(Float64(0.5 * cos(x)), t_1, 1.0) + Float64(0.5 * t_2))) * 0.3333333333333333);
	elseif (x <= 1.05e-5)
		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) / fma(Float64(1.5 * cos(y)), t_2, Float64(fma(0.5, t_1, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(fma(t_0, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_1, cos(x), t_2), 0.5, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $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, -0.0082], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.05e-5], 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[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   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 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right) + 1} \cdot \frac{1}{3} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1} \cdot \frac{1}{3} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3} \]

      11. associate-+r+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   6. Applied rewrites 57.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333 \]

   if -0.00820000000000000069 < x < 1.04999999999999994e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}} \]

   if 1.04999999999999994e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 77.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(t\_1, \cos x, t\_3\right)\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_3, \mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, t\_2, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_4, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma t_1 (cos x) t_3)))
   (if (<= x -0.0082)
     (*
      (/ (fma (* t_2 t_0) (sqrt 2.0) 2.0) (fma 0.5 t_4 1.0))
      0.3333333333333333)
     (if (<= x 1.05e-5)
       (/
        (fma
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         (* (- 1.0 (cos y)) (sqrt 2.0))
         2.0)
        (fma (* 1.5 (cos y)) t_3 (* (fma 0.5 t_1 1.0) 3.0)))
       (/
        (* (fma (* t_0 (sqrt 2.0)) t_2 2.0) 0.3333333333333333)
        (fma t_4 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (0.5 - (cos((x + x)) * 0.5)) * -0.0625;
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(t_1, cos(x), t_3);
	double tmp;
	if (x <= -0.0082) {
		tmp = (fma((t_2 * t_0), sqrt(2.0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.05e-5) {
		tmp = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_3, (fma(0.5, t_1, 1.0) * 3.0));
	} else {
		tmp = (fma((t_0 * sqrt(2.0)), t_2, 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(t_1, cos(x), t_3)
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(fma(Float64(t_2 * t_0), sqrt(2.0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.05e-5)
		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) / fma(Float64(1.5 * cos(y)), t_3, Float64(fma(0.5, t_1, 1.0) * 3.0)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), t_2, 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0));
	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[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[x, -0.0082], N[(N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.05e-5], 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[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$4 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   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 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. 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 57.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -0.00820000000000000069 < x < 1.04999999999999994e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right), 3, \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}} \]

   if 1.04999999999999994e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 77.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(t\_1, \cos x, t\_3\right)\\ \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot t\_0, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, t\_4, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, t\_2, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_4, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma t_1 (cos x) t_3)))
   (if (<= x -0.0082)
     (*
      (/ (fma (* t_2 t_0) (sqrt 2.0) 2.0) (fma 0.5 t_4 1.0))
      0.3333333333333333)
     (if (<= x 1.05e-5)
       (*
        (/
         (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_3 (cos y) t_1) 1.0))
        0.3333333333333333)
       (/
        (* (fma (* t_0 (sqrt 2.0)) t_2 2.0) 0.3333333333333333)
        (fma t_4 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (0.5 - (cos((x + x)) * 0.5)) * -0.0625;
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(t_1, cos(x), t_3);
	double tmp;
	if (x <= -0.0082) {
		tmp = (fma((t_2 * t_0), sqrt(2.0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.05e-5) {
		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_3, cos(y), t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((t_0 * sqrt(2.0)), t_2, 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625)
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(t_1, cos(x), t_3)
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(fma(Float64(t_2 * t_0), sqrt(2.0), 2.0) / fma(0.5, t_4, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.05e-5)
		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_3, cos(y), t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), t_2, 2.0) * 0.3333333333333333) / fma(t_4, 0.5, 1.0));
	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[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[x, -0.0082], N[(N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$4 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.05e-5], 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$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$4 * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   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 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
   5. 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 57.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -0.00820000000000000069 < x < 1.04999999999999994e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   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 98.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 1.04999999999999994e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   6. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 59.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (*
   (fma
    (* (- (cos x) 1.0) (sqrt 2.0))
    (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
    2.0)
   0.3333333333333333)
  (fma (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 0.5 1.0)))

C

double code(double x, double y) {
	return (fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 0.5, 1.0);
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 0.5, 1.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. 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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

6. Applied rewrites 59.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}} \]
7. Add Preprocessing

Alternative 31: 59.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) (- (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((((0.5 - (cos((x + x)) * 0.5)) * -0.0625) * (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(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625) * 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[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $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 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-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 59.3%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625\right) \cdot \left(\cos x - 1\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 32: 42.7% 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 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. 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 42.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 33: 40.2% 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 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. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.2%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(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))))))