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

Percentage Accurate: 99.3% → 99.3%

Time: 9.6s

Alternatives: 31

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

3. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x)))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0), ((1.5 * cos(y)) * (3.0 - sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0), Float64(Float64(1.5 * cos(y)) * Float64(3.0 - sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift--.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.3%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 1\right)}} \]
8. 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: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\\ t_1 := \sin y - 0.0625 \cdot \sin x\\ t_2 := 0.5 \cdot \cos x\\ t_3 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_2, \frac{4}{\sqrt{5} + 1}, 1\right), t\_0\right)}\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_2, \sqrt{5} - 1, 1\right), t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 1.5 (cos y)) (- 3.0 (sqrt 5.0))))
        (t_1 (- (sin y) (* 0.0625 (sin x))))
        (t_2 (* 0.5 (cos x)))
        (t_3
         (/
          (fma (* (sqrt 2.0) (- (cos x) (cos y))) (* (sin x) t_1) 2.0)
          (fma 3.0 (fma t_2 (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0) t_0))))
   (if (<= x -0.021)
     t_3
     (if (<= x 0.0034)
       (/
        (fma
         (* (sqrt 2.0) (- (- 1.0 (* 0.5 (* x x))) (cos y)))
         (* (- (sin x) (* 0.0625 (sin y))) t_1)
         2.0)
        (fma 3.0 (fma t_2 (- (sqrt 5.0) 1.0) 1.0) t_0))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (1.5 * cos(y)) * (3.0 - sqrt(5.0));
	double t_1 = sin(y) - (0.0625 * sin(x));
	double t_2 = 0.5 * cos(x);
	double t_3 = fma((sqrt(2.0) * (cos(x) - cos(y))), (sin(x) * t_1), 2.0) / fma(3.0, fma(t_2, (4.0 / (sqrt(5.0) + 1.0)), 1.0), t_0);
	double tmp;
	if (x <= -0.021) {
		tmp = t_3;
	} else if (x <= 0.0034) {
		tmp = fma((sqrt(2.0) * ((1.0 - (0.5 * (x * x))) - cos(y))), ((sin(x) - (0.0625 * sin(y))) * t_1), 2.0) / fma(3.0, fma(t_2, (sqrt(5.0) - 1.0), 1.0), t_0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.5 * cos(y)) * Float64(3.0 - sqrt(5.0)))
	t_1 = Float64(sin(y) - Float64(0.0625 * sin(x)))
	t_2 = Float64(0.5 * cos(x))
	t_3 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(sin(x) * t_1), 2.0) / fma(3.0, fma(t_2, Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), t_0))
	tmp = 0.0
	if (x <= -0.021)
		tmp = t_3;
	elseif (x <= 0.0034)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y))), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * t_1), 2.0) / fma(3.0, fma(t_2, Float64(sqrt(5.0) - 1.0), 1.0), t_0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.021], t$95$3, If[LessEqual[x, 0.0034], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0210000000000000013 or 0.00339999999999999981 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lift-sin.f64 64.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0210000000000000013 < x < 0.00339999999999999981

   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 inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 99.6

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

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot t\_2}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_0, 0.5 \cdot t\_1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot t\_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (/
          (fma
           (* (sqrt 2.0) t_2)
           (* (sin x) (- (sin y) (* 0.0625 (sin x))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0)))))
   (if (<= x -0.021)
     t_3
     (if (<= x 0.0034)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          t_2))
        (fma
         3.0
         (+ 1.0 (fma 0.5 (* (cos y) t_0) (* 0.5 t_1)))
         (* (* x x) (fma -0.75 t_1 (* 0.0625 (* (* x x) t_1))))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = fma((sqrt(2.0) * t_2), (sin(x) * (sin(y) - (0.0625 * sin(x)))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double tmp;
	if (x <= -0.021) {
		tmp = t_3;
	} else if (x <= 0.0034) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * t_2)) / fma(3.0, (1.0 + fma(0.5, (cos(y) * t_0), (0.5 * t_1))), ((x * x) * fma(-0.75, t_1, (0.0625 * ((x * x) * t_1)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(sin(x) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	tmp = 0.0
	if (x <= -0.021)
		tmp = t_3;
	elseif (x <= 0.0034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * t_2)) / fma(3.0, Float64(1.0 + fma(0.5, Float64(cos(y) * t_0), Float64(0.5 * t_1))), Float64(Float64(x * x) * fma(-0.75, t_1, Float64(0.0625 * Float64(Float64(x * x) * t_1))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.021], t$95$3, If[LessEqual[x, 0.0034], 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[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$1 + N[(0.0625 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0210000000000000013 or 0.00339999999999999981 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lift-sin.f64 64.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0210000000000000013 < x < 0.00339999999999999981

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3, \color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}, {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + \mathsf{fma}\left(0.5, \cos y \cdot \left(3 - \sqrt{5}\right), 0.5 \cdot \left(\sqrt{5} - 1\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, 0.0625 \cdot \left(\left(x \cdot x\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, t\_0, t\_2\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_2, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* (sin x) (- (sin y) (* 0.0625 (sin x))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.021)
     t_1
     (if (<= x 0.0034)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         3.0
         (+ 1.0 (* 0.5 (fma (cos y) t_0 t_2)))
         (* (* x x) (fma -0.75 t_2 (* (* 0.0625 (* x x)) t_2)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (cos(x) - cos(y))), (sin(x) * (sin(y) - (0.0625 * sin(x)))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.021) {
		tmp = t_1;
	} else if (x <= 0.0034) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(3.0, (1.0 + (0.5 * fma(cos(y), t_0, t_2))), ((x * x) * fma(-0.75, t_2, ((0.0625 * (x * x)) * t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(sin(x) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.021)
		tmp = t_1;
	elseif (x <= 0.0034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(3.0, Float64(1.0 + Float64(0.5 * fma(cos(y), t_0, t_2))), Float64(Float64(x * x) * fma(-0.75, t_2, Float64(Float64(0.0625 * Float64(x * x)) * t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.021], t$95$1, If[LessEqual[x, 0.0034], 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[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$2 + N[(N[(0.0625 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0210000000000000013 or 0.00339999999999999981 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lift-sin.f64 64.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 64.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0210000000000000013 < x < 0.00339999999999999981

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, \color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}, {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, t\_0, t\_1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_1, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* (sin x) (- (sin y) (* 0.0625 (sin x))))
           2.0)
          (fma 3.0 (fma (* 0.5 (cos x)) t_1 1.0) (* (* 1.5 (cos y)) t_0)))))
   (if (<= x -0.021)
     t_2
     (if (<= x 0.0034)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         3.0
         (+ 1.0 (* 0.5 (fma (cos y) t_0 t_1)))
         (* (* x x) (fma -0.75 t_1 (* (* 0.0625 (* x x)) t_1)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((sqrt(2.0) * (cos(x) - cos(y))), (sin(x) * (sin(y) - (0.0625 * sin(x)))), 2.0) / fma(3.0, fma((0.5 * cos(x)), t_1, 1.0), ((1.5 * cos(y)) * t_0));
	double tmp;
	if (x <= -0.021) {
		tmp = t_2;
	} else if (x <= 0.0034) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(3.0, (1.0 + (0.5 * fma(cos(y), t_0, t_1))), ((x * x) * fma(-0.75, t_1, ((0.0625 * (x * x)) * t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(sin(x) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), t_1, 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	tmp = 0.0
	if (x <= -0.021)
		tmp = t_2;
	elseif (x <= 0.0034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(3.0, Float64(1.0 + Float64(0.5 * fma(cos(y), t_0, t_1))), Float64(Float64(x * x) * fma(-0.75, t_1, Float64(Float64(0.0625 * Float64(x * x)) * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.021], t$95$2, If[LessEqual[x, 0.0034], 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[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$1 + N[(N[(0.0625 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0210000000000000013 or 0.00339999999999999981 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in y around 0

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

      1. lift-sin.f64 64.4

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

   9. Applied rewrites 64.4%

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

   if -0.0210000000000000013 < x < 0.00339999999999999981

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, \color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}, {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, t\_0, t\_2\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, t\_2, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.021)
     t_1
     (if (<= x 0.0034)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         3.0
         (+ 1.0 (* 0.5 (fma (cos y) t_0 t_2)))
         (* (* x x) (fma -0.75 t_2 (* (* 0.0625 (* x x)) t_2)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (cos(x) - cos(y))), (-0.0625 * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.021) {
		tmp = t_1;
	} else if (x <= 0.0034) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(3.0, (1.0 + (0.5 * fma(cos(y), t_0, t_2))), ((x * x) * fma(-0.75, t_2, ((0.0625 * (x * x)) * t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.021)
		tmp = t_1;
	elseif (x <= 0.0034)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(3.0, Float64(1.0 + Float64(0.5 * fma(cos(y), t_0, t_2))), Float64(Float64(x * x) * fma(-0.75, t_2, Float64(Float64(0.0625 * Float64(x * x)) * t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.021], t$95$1, If[LessEqual[x, 0.0034], 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[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(-0.75 * t$95$2 + N[(N[(0.0625 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0210000000000000013 or 0.00339999999999999981 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

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

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. count-2-rev N/A

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

       8. lower-+.f64 61.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 61.1%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0210000000000000013 < x < 0.00339999999999999981

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(3, \color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}, {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3, 1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(-0.75, \sqrt{5} - 1, \left(0.0625 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.0115:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_3\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           (* (sqrt 2.0) t_1)
           (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.0115)
     t_2
     (if (<= x 0.0033)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma
             x
             (- 1.0 (* 0.16666666666666666 (* x x)))
             (* -0.0625 (sin y))))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          t_1))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_3) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_3)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma((sqrt(2.0) * t_1), (-0.0625 * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.0115) {
		tmp = t_2;
	} else if (x <= 0.0033) {
		tmp = (2.0 + (((sqrt(2.0) * fma(x, (1.0 - (0.16666666666666666 * (x * x))), (-0.0625 * sin(y)))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, ((-0.75 * (x * x)) * t_3));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.0115)
		tmp = t_2;
	elseif (x <= 0.0033)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(x, Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))), Float64(-0.0625 * sin(y)))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_3)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0115], t$95$2, If[LessEqual[x, 0.0033], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0115 or 0.0033 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

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

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. count-2-rev N/A

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

       8. lower-+.f64 61.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 61.1%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0115 < x < 0.0033

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, \color{blue}{1 + \frac{-1}{6} \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot {\color{blue}{x}}^{2}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \color{blue}{{x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot x\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lift-sin.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.0115:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.0115)
     t_1
     (if (<= x 0.0033)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma
             x
             (- 1.0 (* 0.16666666666666666 (* x x)))
             (* -0.0625 (sin y))))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (cos(x) - cos(y))), (-0.0625 * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.0115) {
		tmp = t_1;
	} else if (x <= 0.0033) {
		tmp = (2.0 + (((sqrt(2.0) * fma(x, (1.0 - (0.16666666666666666 * (x * x))), (-0.0625 * sin(y)))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.0115)
		tmp = t_1;
	elseif (x <= 0.0033)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(x, Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))), Float64(-0.0625 * sin(y)))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0115], t$95$1, If[LessEqual[x, 0.0033], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0115 or 0.0033 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

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

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. count-2-rev N/A

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

       8. lower-+.f64 61.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 61.1%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.0115 < x < 0.0033

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   12. Step-by-step derivation

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

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) + \frac{-1}{16} \cdot \sin \color{blue}{y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, \color{blue}{1 + \frac{-1}{6} \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

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

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot {\color{blue}{x}}^{2}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \color{blue}{{x}^{2}}, \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       8. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot \color{blue}{x}\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - \frac{1}{6} \cdot \left(x \cdot x\right), \frac{-1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       11. lift-sin.f64 99.6

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   13. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.011:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- (cos x) (cos y)))
           (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
           2.0)
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.011)
     t_1
     (if (<= x 0.0033)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (cos(x) - cos(y))), (-0.0625 * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.011) {
		tmp = t_1;
	} else if (x <= 0.0033) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.011)
		tmp = t_1;
	elseif (x <= 0.0033)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.011], t$95$1, If[LessEqual[x, 0.0033], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010999999999999999 or 0.0033 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

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

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. count-2-rev N/A

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

       8. lower-+.f64 61.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 61.1%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.010999999999999999 < x < 0.0033

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \color{blue}{\frac{1}{16} \cdot \sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       3. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \color{blue}{\sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 0.5 \cdot \cos x\\ t_2 := \sqrt{5} - 1\\ t_3 := \left(1.5 \cdot \cos y\right) \cdot t\_0\\ \mathbf{if}\;x \leq -0.011:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, \frac{4}{\sqrt{5} + 1}, 1\right), t\_3\right)}\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, t\_2, 1\right), t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* 0.5 (cos x)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* (* 1.5 (cos y)) t_0)))
   (if (<= x -0.011)
     (/
      (-
       2.0
       (*
        0.0625
        (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
      (fma 3.0 (fma t_1 (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0) t_3))
     (if (<= x 0.0033)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       (/
        (fma
         (* (sqrt 2.0) (- (cos x) (cos y)))
         (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         2.0)
        (fma 3.0 (fma t_1 t_2 1.0) t_3))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 0.5 * cos(x);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = (1.5 * cos(y)) * t_0;
	double tmp;
	if (x <= -0.011) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma(t_1, (4.0 / (sqrt(5.0) + 1.0)), 1.0), t_3);
	} else if (x <= 0.0033) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = fma((sqrt(2.0) * (cos(x) - cos(y))), (-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), 2.0) / fma(3.0, fma(t_1, t_2, 1.0), t_3);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(0.5 * cos(x))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(1.5 * cos(y)) * t_0)
	tmp = 0.0
	if (x <= -0.011)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(t_1, Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), t_3));
	elseif (x <= 0.0033)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), 2.0) / fma(3.0, fma(t_1, t_2, 1.0), t_3));
	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[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.011], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0033], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 * t$95$2 + 1.0), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.010999999999999999

   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 99.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

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

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lower--.f64 N/A

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

   11. Applied rewrites 61.2%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{3}, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.010999999999999999 < x < 0.0033

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \color{blue}{\frac{1}{16} \cdot \sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       3. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \color{blue}{\sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   if 0.0033 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

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

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 60.9

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

   9. Applied rewrites 60.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.011:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (-
           2.0
           (*
            0.0625
            (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (fma
           3.0
           (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
           (* (* 1.5 (cos y)) t_0))))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.011)
     t_1
     (if (<= x 0.0033)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
           (- (sin y) (* x (fma -0.010416666666666666 (* x x) 0.0625))))
          (- (- 1.0 (* 0.5 (* x x))) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.011) {
		tmp = t_1;
	} else if (x <= 0.0033) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (x * fma(-0.010416666666666666, (x * x), 0.0625)))) * ((1.0 - (0.5 * (x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.011)
		tmp = t_1;
	elseif (x <= 0.0033)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(x * fma(-0.010416666666666666, Float64(x * x), 0.0625)))) * Float64(Float64(1.0 - Float64(0.5 * Float64(x * x))) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	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[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.011], t$95$1, If[LessEqual[x, 0.0033], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x * N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.010999999999999999 or 0.0033 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

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

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lower--.f64 N/A

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

   11. Applied rewrites 61.1%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{3}, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.010999999999999999 < x < 0.0033

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{96} \cdot {x}^{2}\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \left(\frac{-1}{96} \cdot {x}^{2} + \color{blue}{\frac{1}{16}}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, \color{blue}{{x}^{2}}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot \color{blue}{x}, \frac{1}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot \color{blue}{x}, 0.0625\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot {x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot {\color{blue}{x}}^{2}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \color{blue}{{x}^{2}}\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lift-*.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot \color{blue}{x}\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\color{blue}{\left(1 - 0.5 \cdot \left(x \cdot x\right)\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \color{blue}{\frac{1}{16} \cdot \sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       2. lift-sin.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{1}{16} \cdot \sin y\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{1}{16}\right)\right)\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(\frac{-3}{4} \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

       3. lift-*.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \color{blue}{\sin y}\right)\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - x \cdot \mathsf{fma}\left(-0.010416666666666666, x \cdot x, 0.0625\right)\right)\right) \cdot \left(\left(1 - 0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{5} - 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{2} \cdot \left(1 - \cos y\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_3\right), 1\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_1 \cdot t\_2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_3\right)}\\ \mathbf{elif}\;y \leq 0.0047:\\ \;\;\;\;\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}{t\_4} + \frac{2}{t\_4}\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right), 3, \left(\cos y \cdot \frac{t\_3}{2}\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (* (sqrt 2.0) (- 1.0 (cos y))))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma 0.5 (fma t_0 (cos x) t_3) 1.0)))
   (if (<= y -6.5e-5)
     (/
      (- 2.0 (* 0.0625 (* t_1 t_2)))
      (fma 3.0 (fma (* 0.5 (cos x)) t_0 1.0) (* (* 1.5 (cos y)) t_3)))
     (if (<= y 0.0047)
       (*
        (+
         (/
          (*
           (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
           (* (- (cos x) 1.0) (sqrt 2.0)))
          t_4)
         (/ 2.0 t_4))
        0.3333333333333333)
       (/
        (fma (* -0.0625 t_1) t_2 2.0)
        (fma
         (fma (cos x) (/ t_0 2.0) 1.0)
         3.0
         (* (* (cos y) (/ t_3 2.0)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(2.0) * (1.0 - cos(y));
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(0.5, fma(t_0, cos(x), t_3), 1.0);
	double tmp;
	if (y <= -6.5e-5) {
		tmp = (2.0 - (0.0625 * (t_1 * t_2))) / fma(3.0, fma((0.5 * cos(x)), t_0, 1.0), ((1.5 * cos(y)) * t_3));
	} else if (y <= 0.0047) {
		tmp = ((((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * ((cos(x) - 1.0) * sqrt(2.0))) / t_4) + (2.0 / t_4)) * 0.3333333333333333;
	} else {
		tmp = fma((-0.0625 * t_1), t_2, 2.0) / fma(fma(cos(x), (t_0 / 2.0), 1.0), 3.0, ((cos(y) * (t_3 / 2.0)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(2.0) * Float64(1.0 - cos(y)))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(0.5, fma(t_0, cos(x), t_3), 1.0)
	tmp = 0.0
	if (y <= -6.5e-5)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_1 * t_2))) / fma(3.0, fma(Float64(0.5 * cos(x)), t_0, 1.0), Float64(Float64(1.5 * cos(y)) * t_3)));
	elseif (y <= 0.0047)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) / t_4) + Float64(2.0 / t_4)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(-0.0625 * t_1), t_2, 2.0) / fma(fma(cos(x), Float64(t_0 / 2.0), 1.0), 3.0, Float64(Float64(cos(y) * Float64(t_3 / 2.0)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -6.5e-5], N[(N[(2.0 - N[(0.0625 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0047], N[(N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(2.0 / t$95$4), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.49999999999999943e-5

   1. Initial program 99.0%

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

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-cos.f64 60.8

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

   9. Applied rewrites 60.8%

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

   if -6.49999999999999943e-5 < y < 0.00470000000000000018

   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 98.6%

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

   6. Applied rewrites 98.7%

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

   if 0.00470000000000000018 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      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 99.2%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.0

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

   6. Applied rewrites 60.0%

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

Alternative 17: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 0.5 \cdot \cos x\\ t_2 := \sqrt{5} - 1\\ t_3 := {\sin y}^{2}\\ t_4 := \left(1.5 \cdot \cos y\right) \cdot t\_0\\ t_5 := \sqrt{2} \cdot \left(1 - \cos y\right)\\ \mathbf{if}\;y \leq -0.00086:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot t\_5\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, t\_2, 1\right), t\_4\right)}\\ \mathbf{elif}\;y \leq 0.0047:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, \frac{4}{\sqrt{5} + 1}, 1\right), t\_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_5, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right), 3, \left(\cos y \cdot \frac{t\_0}{2}\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* 0.5 (cos x)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (pow (sin y) 2.0))
        (t_4 (* (* 1.5 (cos y)) t_0))
        (t_5 (* (sqrt 2.0) (- 1.0 (cos y)))))
   (if (<= y -0.00086)
     (/ (- 2.0 (* 0.0625 (* t_3 t_5))) (fma 3.0 (fma t_1 t_2 1.0) t_4))
     (if (<= y 0.0047)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (fma t_1 (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0) t_4))
       (/
        (fma (* -0.0625 t_3) t_5 2.0)
        (fma
         (fma (cos x) (/ t_2 2.0) 1.0)
         3.0
         (* (* (cos y) (/ t_0 2.0)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 0.5 * cos(x);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = pow(sin(y), 2.0);
	double t_4 = (1.5 * cos(y)) * t_0;
	double t_5 = sqrt(2.0) * (1.0 - cos(y));
	double tmp;
	if (y <= -0.00086) {
		tmp = (2.0 - (0.0625 * (t_3 * t_5))) / fma(3.0, fma(t_1, t_2, 1.0), t_4);
	} else if (y <= 0.0047) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma(t_1, (4.0 / (sqrt(5.0) + 1.0)), 1.0), t_4);
	} else {
		tmp = fma((-0.0625 * t_3), t_5, 2.0) / fma(fma(cos(x), (t_2 / 2.0), 1.0), 3.0, ((cos(y) * (t_0 / 2.0)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(0.5 * cos(x))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = sin(y) ^ 2.0
	t_4 = Float64(Float64(1.5 * cos(y)) * t_0)
	t_5 = Float64(sqrt(2.0) * Float64(1.0 - cos(y)))
	tmp = 0.0
	if (y <= -0.00086)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * t_5))) / fma(3.0, fma(t_1, t_2, 1.0), t_4));
	elseif (y <= 0.0047)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(t_1, Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), t_4));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_3), t_5, 2.0) / fma(fma(cos(x), Float64(t_2 / 2.0), 1.0), 3.0, Float64(Float64(cos(y) * Float64(t_0 / 2.0)) * 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[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00086], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * t$95$2 + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0047], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * t$95$5 + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.59999999999999979e-4

   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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-cos.f64 60.8

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

   9. Applied rewrites 60.8%

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

   if -8.59999999999999979e-4 < y < 0.00470000000000000018

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

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

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lower--.f64 N/A

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

   11. Applied rewrites 98.8%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{3}, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if 0.00470000000000000018 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      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 99.2%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.0

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

   6. Applied rewrites 60.0%

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

Alternative 18: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\\ t_1 := 0.5 \cdot \cos x\\ t_2 := \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, \sqrt{5} - 1, 1\right), t\_0\right)}\\ \mathbf{if}\;y \leq -0.00086:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0047:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(t\_1, \frac{4}{\sqrt{5} + 1}, 1\right), t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 1.5 (cos y)) (- 3.0 (sqrt 5.0))))
        (t_1 (* 0.5 (cos x)))
        (t_2
         (/
          (-
           2.0
           (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
          (fma 3.0 (fma t_1 (- (sqrt 5.0) 1.0) 1.0) t_0))))
   (if (<= y -0.00086)
     t_2
     (if (<= y 0.0047)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 3.0 (fma t_1 (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0) t_0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = (1.5 * cos(y)) * (3.0 - sqrt(5.0));
	double t_1 = 0.5 * cos(x);
	double t_2 = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(3.0, fma(t_1, (sqrt(5.0) - 1.0), 1.0), t_0);
	double tmp;
	if (y <= -0.00086) {
		tmp = t_2;
	} else if (y <= 0.0047) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma(t_1, (4.0 / (sqrt(5.0) + 1.0)), 1.0), t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(1.5 * cos(y)) * Float64(3.0 - sqrt(5.0)))
	t_1 = Float64(0.5 * cos(x))
	t_2 = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(3.0, fma(t_1, Float64(sqrt(5.0) - 1.0), 1.0), t_0))
	tmp = 0.0
	if (y <= -0.00086)
		tmp = t_2;
	elseif (y <= 0.0047)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(t_1, Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), t_0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00086], t$95$2, If[LessEqual[y, 0.0047], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.59999999999999979e-4 or 0.00470000000000000018 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-cos.f64 60.4

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

   9. Applied rewrites 60.4%

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

   if -8.59999999999999979e-4 < y < 0.00470000000000000018

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

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

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lower--.f64 N/A

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

   11. Applied rewrites 98.8%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{3}, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.00086:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0047:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
            (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -0.00086)
     t_1
     (if (<= y 0.0047)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma
         3.0
         (fma (* 0.5 (cos x)) (/ 4.0 (+ (sqrt 5.0) 1.0)) 1.0)
         (* (* 1.5 (cos y)) t_0)))
       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) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -0.00086) {
		tmp = t_1;
	} else if (y <= 0.0047) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma((0.5 * cos(x)), (4.0 / (sqrt(5.0) + 1.0)), 1.0), ((1.5 * cos(y)) * t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -0.00086)
		tmp = t_1;
	elseif (y <= 0.0047)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(Float64(0.5 * cos(x)), Float64(4.0 / Float64(sqrt(5.0) + 1.0)), 1.0), Float64(Float64(1.5 * cos(y)) * t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(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[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00086], t$95$1, If[LessEqual[y, 0.0047], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.59999999999999979e-4 or 0.00470000000000000018 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

   if -8.59999999999999979e-4 < y < 0.00470000000000000018

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

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

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-cos.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-sqrt.f64 N/A

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

       15. lower--.f64 N/A

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

   11. Applied rewrites 98.8%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\color{blue}{3}, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\sqrt{5} + 1}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0047:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1.5, t\_0, 3 \cdot \left(1 - -2 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
            (* (/ t_0 2.0) (cos y)))))))
   (if (<= y -6.5e-5)
     t_1
     (if (<= y 0.0047)
       (/
        (-
         2.0
         (*
          0.0625
          (* (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (fma 1.5 t_0 (* 3.0 (- 1.0 (* -2.0 (/ (cos x) (+ 1.0 (sqrt 5.0))))))))
       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) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (y <= -6.5e-5) {
		tmp = t_1;
	} else if (y <= 0.0047) {
		tmp = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((x + x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(1.5, t_0, (3.0 * (1.0 - (-2.0 * (cos(x) / (1.0 + sqrt(5.0)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -6.5e-5)
		tmp = t_1;
	elseif (y <= 0.0047)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(1.5, t_0, Float64(3.0 * Float64(1.0 - Float64(-2.0 * Float64(cos(x) / Float64(1.0 + sqrt(5.0))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(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[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e-5], t$95$1, If[LessEqual[y, 0.0047], N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$0 + N[(3.0 * N[(1.0 - N[(-2.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999943e-5 or 0.00470000000000000018 < y

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

   if -6.49999999999999943e-5 < y < 0.00470000000000000018

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      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 99.5%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 98.7%

       \[\leadsto \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3 \cdot \left(1 - -2 \cdot \frac{\cos x}{1 + \sqrt{5}}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{2 - 0.0625 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(1 - -0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (-
           2.0
           (*
            0.0625
            (*
             (- 0.5 (* 0.5 (cos (* 2.0 x))))
             (* (sqrt 2.0) (- (cos x) 1.0)))))
          (fma 3.0 (fma (* 0.5 (cos x)) t_1 1.0) (* (* 1.5 (cos y)) t_0)))))
   (if (<= x -6.2e-6)
     t_2
     (if (<= x 0.0005)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (- 1.0 (* -0.5 t_1)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (2.0 - (0.0625 * ((0.5 - (0.5 * cos((2.0 * x)))) * (sqrt(2.0) * (cos(x) - 1.0))))) / fma(3.0, fma((0.5 * cos(x)), t_1, 1.0), ((1.5 * cos(y)) * t_0));
	double tmp;
	if (x <= -6.2e-6) {
		tmp = t_2;
	} else if (x <= 0.0005) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (1.0 - (-0.5 * t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(2.0 - Float64(0.0625 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x)))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / fma(3.0, fma(Float64(0.5 * cos(x)), t_1, 1.0), Float64(Float64(1.5 * cos(y)) * t_0)))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = t_2;
	elseif (x <= 0.0005)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 - N[(0.0625 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], t$95$2, If[LessEqual[x, 0.0005], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999999e-6 or 5.0000000000000001e-4 < 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. 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 x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in y around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 61.0%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 - -0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{\mathsf{fma}\left(1.5, t\_2, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_2, 3 \cdot \left(1 - -0.5 \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_1 \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 0.5 (* 0.5 (cos (+ x x))))))
   (if (<= x -6.2e-6)
     (/
      (- 2.0 (* 0.0625 (* t_3 (* (sqrt 2.0) t_1))))
      (fma 1.5 t_2 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_0))))))
     (if (<= x 0.0005)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_2) (* 3.0 (- 1.0 (* -0.5 t_0)))))
       (/
        (* (fma (* -0.0625 t_3) (* t_1 (sqrt 2.0)) 2.0) 0.3333333333333333)
        (fma 0.5 (fma t_0 (cos x) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 - (0.5 * cos((x + x)));
	double tmp;
	if (x <= -6.2e-6) {
		tmp = (2.0 - (0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / fma(1.5, t_2, (3.0 * (1.0 - (-0.5 * (cos(x) * t_0)))));
	} else if (x <= 0.0005) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_2), (3.0 * (1.0 - (-0.5 * t_0))));
	} else {
		tmp = (fma((-0.0625 * t_3), (t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_1)))) / fma(1.5, t_2, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_0))))));
	elseif (x <= 0.0005)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_2), Float64(3.0 * Float64(1.0 - Float64(-0.5 * t_0)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), Float64(t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$2 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0005], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(3.0 * N[(1.0 - N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 99.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 99.2%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 23: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{\mathsf{fma}\left(1.5, t\_2, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_2, 3 \cdot \mathsf{fma}\left(0.5, t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_1 \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 0.5 (* 0.5 (cos (+ x x))))))
   (if (<= x -6.2e-6)
     (/
      (- 2.0 (* 0.0625 (* t_3 (* (sqrt 2.0) t_1))))
      (fma 1.5 t_2 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_0))))))
     (if (<= x 0.0005)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma 1.5 (* (cos y) t_2) (* 3.0 (fma 0.5 t_0 1.0))))
       (/
        (* (fma (* -0.0625 t_3) (* t_1 (sqrt 2.0)) 2.0) 0.3333333333333333)
        (fma 0.5 (fma t_0 (cos x) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 - (0.5 * cos((x + x)));
	double tmp;
	if (x <= -6.2e-6) {
		tmp = (2.0 - (0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / fma(1.5, t_2, (3.0 * (1.0 - (-0.5 * (cos(x) * t_0)))));
	} else if (x <= 0.0005) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma(1.5, (cos(y) * t_2), (3.0 * fma(0.5, t_0, 1.0)));
	} else {
		tmp = (fma((-0.0625 * t_3), (t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_1)))) / fma(1.5, t_2, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_0))))));
	elseif (x <= 0.0005)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(1.5, Float64(cos(y) * t_2), Float64(3.0 * fma(0.5, t_0, 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), Float64(t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$2 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0005], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(3.0 * N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 99.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift--.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 24: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{\mathsf{fma}\left(1.5, t\_2, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, 3 \cdot \mathsf{fma}\left(0.5, t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_1 \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 0.5 (* 0.5 (cos (+ x x))))))
   (if (<= x -6.2e-6)
     (/
      (- 2.0 (* 0.0625 (* t_3 (* (sqrt 2.0) t_1))))
      (fma 1.5 t_2 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_0))))))
     (if (<= x 0.0005)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_2 (* 3.0 (fma 0.5 t_0 1.0))))
       (/
        (* (fma (* -0.0625 t_3) (* t_1 (sqrt 2.0)) 2.0) 0.3333333333333333)
        (fma 0.5 (fma t_0 (cos x) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 - (0.5 * cos((x + x)));
	double tmp;
	if (x <= -6.2e-6) {
		tmp = (2.0 - (0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / fma(1.5, t_2, (3.0 * (1.0 - (-0.5 * (cos(x) * t_0)))));
	} else if (x <= 0.0005) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_2, (3.0 * fma(0.5, t_0, 1.0)));
	} else {
		tmp = (fma((-0.0625 * t_3), (t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_1)))) / fma(1.5, t_2, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_0))))));
	elseif (x <= 0.0005)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_2, Float64(3.0 * fma(0.5, t_0, 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), Float64(t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$2 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0005], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(3.0 * N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 99.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.2%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 25: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{\mathsf{fma}\left(1.5, t\_2, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_3, t\_1 \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (- 0.5 (* 0.5 (cos (+ x x))))))
   (if (<= x -6.2e-6)
     (/
      (- 2.0 (* 0.0625 (* t_3 (* (sqrt 2.0) t_1))))
      (fma 1.5 t_2 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_0))))))
     (if (<= x 0.0005)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        0.3333333333333333)
       (/
        (* (fma (* -0.0625 t_3) (* t_1 (sqrt 2.0)) 2.0) 0.3333333333333333)
        (fma 0.5 (fma t_0 (cos x) t_2) 1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 - (0.5 * cos((x + x)));
	double tmp;
	if (x <= -6.2e-6) {
		tmp = (2.0 - (0.0625 * (t_3 * (sqrt(2.0) * t_1)))) / fma(1.5, t_2, (3.0 * (1.0 - (-0.5 * (cos(x) * t_0)))));
	} else if (x <= 0.0005) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * t_3), (t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_1)))) / fma(1.5, t_2, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_0))))));
	elseif (x <= 0.0005)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_3), Float64(t_1 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_0, cos(x), t_2), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(2.0 - N[(0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$2 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0005], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$3), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 99.0%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, \frac{4}{\color{blue}{\sqrt{5} + 1}}, 1\right), \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 26: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - 1\\ t_1 := -0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot t\_0, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 1.5 \cdot t\_3\right)}\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0 \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_3\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) 1.0))
        (t_1 (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x))))))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -6.2e-6)
     (/
      (fma t_1 (* (sqrt 2.0) t_0) 2.0)
      (fma 3.0 (fma (* 0.5 (cos x)) t_2 1.0) (* 1.5 t_3)))
     (if (<= x 0.0005)
       (*
        (/
         (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_2) 1.0))
        0.3333333333333333)
       (/
        (* (fma t_1 (* t_0 (sqrt 2.0)) 2.0) 0.3333333333333333)
        (fma 0.5 (fma t_2 (cos x) t_3) 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = -0.0625 * (0.5 - (0.5 * cos((x + x))));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -6.2e-6) {
		tmp = fma(t_1, (sqrt(2.0) * t_0), 2.0) / fma(3.0, fma((0.5 * cos(x)), t_2, 1.0), (1.5 * t_3));
	} else if (x <= 0.0005) {
		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_2), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma(t_1, (t_0 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), t_3), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * t_0), 2.0) / fma(3.0, fma(Float64(0.5 * cos(x)), t_2, 1.0), Float64(1.5 * t_3)));
	elseif (x <= 0.0005)
		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_2), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_1, Float64(t_0 * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(0.5, fma(t_2, cos(x), t_3), 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[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] + N[(1.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0005], 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$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$1 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 99.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 27: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 - 0.5 \cdot \cos \left(x + x\right)\\ t_3 := 3 - \sqrt{5}\\ t_4 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_3\right), 1\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_2 \cdot t\_0, 2\right)}{t\_4} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_0, 2\right) \cdot 0.3333333333333333}{t\_4}\\ \end{array} \end{array} \]

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 (- 0.5 (* 0.5 (cos (+ x x)))))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (fma 0.5 (fma t_1 (cos x) t_3) 1.0)))
   (if (<= x -6.2e-6)
     (* (/ (fma -0.0625 (* t_2 t_0) 2.0) t_4) 0.3333333333333333)
     (if (<= x 0.0005)
       (*
        (/
         (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 (* -0.0625 t_2) t_0 2.0) 0.3333333333333333) t_4)))))

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 = 0.5 - (0.5 * cos((x + x)));
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = fma(0.5, fma(t_1, cos(x), t_3), 1.0);
	double tmp;
	if (x <= -6.2e-6) {
		tmp = (fma(-0.0625, (t_2 * t_0), 2.0) / t_4) * 0.3333333333333333;
	} else if (x <= 0.0005) {
		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((-0.0625 * t_2), t_0, 2.0) * 0.3333333333333333) / t_4;
	}
	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(0.5 - Float64(0.5 * cos(Float64(x + x))))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = fma(0.5, fma(t_1, cos(x), t_3), 1.0)
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = Float64(Float64(fma(-0.0625, Float64(t_2 * t_0), 2.0) / t_4) * 0.3333333333333333);
	elseif (x <= 0.0005)
		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(-0.0625 * t_2), t_0, 2.0) * 0.3333333333333333) / t_4);
	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[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e-6], N[(N[(N[(-0.0625 * N[(t$95$2 * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 0.0005], 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[(-0.0625 * t$95$2), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 60.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   6. Applied rewrites 60.1%

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

   if -6.1999999999999999e-6 < x < 5.0000000000000001e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.7%

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

   6. Applied rewrites 59.8%

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

Alternative 28: 60.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.3%

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

6. Applied rewrites 60.3%

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

Alternative 29: 60.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.3%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 60.3%

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

Alternative 30: 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 60.3%

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

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 31: 40.0% accurate, 316.7× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 60.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.0%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(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))))))