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

Percentage Accurate: 99.3% → 99.3%

Time: 12.4s

Alternatives: 28

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (cos x) (cos y)))
   (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))
   2.0)
  (fma
   (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) 1.0)
   3.0
   (* (* 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(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0), 3.0, ((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(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0), 3.0, 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[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(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. Add Preprocessing
3. 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

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (cos x) (cos y)))
   (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))
   2.0)
  (fma
   (fma 0.5 (* (cos x) (- (sqrt 5.0) 1.0)) 1.0)
   3.0
   (* (* 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(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(fma(0.5, (cos(x) * (sqrt(5.0) - 1.0)), 1.0), 3.0, ((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(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(fma(0.5, Float64(cos(x) * Float64(sqrt(5.0) - 1.0)), 1.0), 3.0, 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[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + 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. Add Preprocessing
3. 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

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift--.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (* (sqrt 2.0) (- (cos x) (cos y)))
    (* (- (sin y) (* 0.0625 (sin x))) (- (sin x) (* 0.0625 (sin y))))
    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((sqrt(2.0) * (cos(x) - cos(y))), ((sin(y) - (0.0625 * sin(x))) * (sin(x) - (0.0625 * sin(y)))), 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(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 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[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 99.2%

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

Alternative 4: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{2} \cdot t\_1\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -3:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_3, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.78:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.0005208333333333333 + -1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot y\right)\right) - 0.010416666666666666\right)\right)\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_0 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_3, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (* (sqrt 2.0) t_1))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= y -3.0)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_3) (* 0.5 (* (cos y) t_0))))))
     (if (<= y 0.78)
       (*
        (/
         (fma
          t_2
          (*
           (- (sin y) (* 0.0625 (sin x)))
           (-
            (sin x)
            (*
             y
             (+
              0.0625
              (*
               (* y y)
               (-
                (*
                 (* y y)
                 (+ 0.0005208333333333333 (* -1.240079365079365e-5 (* y y))))
                0.010416666666666666))))))
          2.0)
         (fma 0.5 (fma t_3 (cos x) (* t_0 (cos y))) 1.0))
        0.3333333333333333)
       (/
        (fma t_2 (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
        (fma (fma (* 0.5 (cos x)) t_3 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))))))

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 = sqrt(2.0) * t_1;
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -3.0) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_3), (0.5 * (cos(y) * t_0)))));
	} else if (y <= 0.78) {
		tmp = (fma(t_2, ((sin(y) - (0.0625 * sin(x))) * (sin(x) - (y * (0.0625 + ((y * y) * (((y * y) * (0.0005208333333333333 + (-1.240079365079365e-5 * (y * y)))) - 0.010416666666666666)))))), 2.0) / fma(0.5, fma(t_3, cos(x), (t_0 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(t_2, (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(fma((0.5 * cos(x)), t_3, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	}
	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(sqrt(2.0) * t_1)
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -3.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_3), Float64(0.5 * Float64(cos(y) * t_0))))));
	elseif (y <= 0.78)
		tmp = Float64(Float64(fma(t_2, Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) - Float64(y * Float64(0.0625 + Float64(Float64(y * y) * Float64(Float64(Float64(y * y) * Float64(0.0005208333333333333 + Float64(-1.240079365079365e-5 * Float64(y * y)))) - 0.010416666666666666)))))), 2.0) / fma(0.5, fma(t_3, cos(x), Float64(t_0 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(t_2, Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_3, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -3.0], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.78], N[(N[(N[(t$95$2 * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y * N[(0.0625 + N[(N[(y * y), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * N[(0.0005208333333333333 + N[(-1.240079365079365e-5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 23.1

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 22.9

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

   8. Applied rewrites 22.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift--.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -3 < y < 0.78000000000000003

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.0005208333333333333 + -1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot y\right)\right) - 0.010416666666666666\right)\right)\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - y \cdot \left(0.0625 + \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.0005208333333333333 + -1.240079365079365 \cdot 10^{-5} \cdot \left(y \cdot y\right)\right) - 0.010416666666666666\right)\right)\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   if 0.78000000000000003 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 65.5

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

   10. Applied rewrites 65.5%

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

Alternative 5: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.47:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos x \cdot t\_2, 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.245:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_2, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -0.47)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (* 3.0 (+ 1.0 (fma 0.5 (* (cos x) t_2) (* 0.5 (* (cos y) t_0))))))
     (if (<= y 0.245)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y)))
           (fma
            (fma
             (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
             (* y y)
             1.0)
            y
            (* -0.0625 (sin x))))
          t_1))
        (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
       (/
        (fma (* (sqrt 2.0) t_1) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
        (fma (fma (* 0.5 (cos x)) t_2 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))))))

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 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.47) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(x) * t_2), (0.5 * (cos(y) * t_0)))));
	} else if (y <= 0.245) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	} else {
		tmp = fma((sqrt(2.0) * t_1), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(fma((0.5 * cos(x)), t_2, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	}
	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(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.47)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(x) * t_2), Float64(0.5 * Float64(cos(y) * t_0))))));
	elseif (y <= 0.245)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_2, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.47], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.245], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 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], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.46999999999999997

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 23.0

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

   5. Applied rewrites 23.0%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 22.8

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

   8. Applied rewrites 22.8%

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

   9. Taylor expanded in x around inf

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-cos.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift--.f64 59.1

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

   11. Applied rewrites 59.1%

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

   if -0.46999999999999997 < y < 0.245

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.245 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 65.5

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

   10. Applied rewrites 65.5%

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

Alternative 6: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.47 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= y -0.47) (not (<= y 0.245)))
     (/
      (fma (* (sqrt 2.0) t_2) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
      (fma (fma (* 0.5 (cos x)) t_1 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma
          (fma
           (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
           (* y y)
           1.0)
          y
          (* -0.0625 (sin x))))
        t_2))
      (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((y <= -0.47) || !(y <= 0.245)) {
		tmp = fma((sqrt(2.0) * t_2), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(fma((0.5 * cos(x)), t_1, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0), y, (-0.0625 * sin(x)))) * t_2)) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((y <= -0.47) || !(y <= 0.245))
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_1, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0), y, Float64(-0.0625 * sin(x)))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.47], N[Not[LessEqual[y, 0.245]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.46999999999999997 or 0.245 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 62.5

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

   10. Applied rewrites 62.5%

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

   if -0.46999999999999997 < y < 0.245

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.47 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.47 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -0.47) (not (<= y 0.245)))
     (*
      (/
       (fma (* (sqrt 2.0) t_0) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
       (fma 0.5 (fma t_2 (cos x) (* t_1 (cos y))) 1.0))
      0.3333333333333333)
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma
          (fma
           (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
           (* y y)
           1.0)
          y
          (* -0.0625 (sin x))))
        t_0))
      (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_1 2.0) (cos y))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -0.47) || !(y <= 0.245)) {
		tmp = (fma((sqrt(2.0) * t_0), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma(t_2, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0), y, (-0.0625 * sin(x)))) * t_0)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -0.47) || !(y <= 0.245))
		tmp = Float64(Float64(fma(Float64(sqrt(2.0) * t_0), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(t_2, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0), y, Float64(-0.0625 * sin(x)))) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.47], N[Not[LessEqual[y, 0.245]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.46999999999999997 or 0.245 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if -0.46999999999999997 < y < 0.245

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.47 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ t_4 := \frac{t\_3}{2}\\ t_5 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.072:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\cos y, t\_4, \mathsf{fma}\left(\cos x, t\_1, 1\right)\right) \cdot 3} + \frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot t\_5}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_0\right), 1\right)} \cdot -0.020833333333333332\\ \mathbf{elif}\;y \leq 0.245:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_4 \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, -0.0625 \cdot t\_5, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0)))
        (t_4 (/ t_3 2.0))
        (t_5 (pow (sin y) 2.0)))
   (if (<= y -0.072)
     (+
      (/ 2.0 (* (fma (cos y) t_4 (fma (cos x) t_1 1.0)) 3.0))
      (*
       (/
        (* (* (- 1.0 (cos y)) (sqrt 2.0)) t_5)
        (fma 0.5 (fma t_3 (cos y) t_0) 1.0))
       -0.020833333333333332))
     (if (<= y 0.245)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y)))
           (fma
            (fma
             (- (* (* y y) 0.008333333333333333) 0.16666666666666666)
             (* y y)
             1.0)
            y
            (* -0.0625 (sin x))))
          t_2))
        (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_4 (cos y)))))
       (/
        (fma (* (sqrt 2.0) t_2) (* -0.0625 t_5) 2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_3)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double t_4 = t_3 / 2.0;
	double t_5 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.072) {
		tmp = (2.0 / (fma(cos(y), t_4, fma(cos(x), t_1, 1.0)) * 3.0)) + (((((1.0 - cos(y)) * sqrt(2.0)) * t_5) / fma(0.5, fma(t_3, cos(y), t_0), 1.0)) * -0.020833333333333332);
	} else if (y <= 0.245) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((((y * y) * 0.008333333333333333) - 0.16666666666666666), (y * y), 1.0), y, (-0.0625 * sin(x)))) * t_2)) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_4 * cos(y))));
	} else {
		tmp = fma((sqrt(2.0) * t_2), (-0.0625 * t_5), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_3));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	t_4 = Float64(t_3 / 2.0)
	t_5 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.072)
		tmp = Float64(Float64(2.0 / Float64(fma(cos(y), t_4, fma(cos(x), t_1, 1.0)) * 3.0)) + Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * t_5) / fma(0.5, fma(t_3, cos(y), t_0), 1.0)) * -0.020833333333333332));
	elseif (y <= 0.245)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(Float64(Float64(y * y) * 0.008333333333333333) - 0.16666666666666666), Float64(y * y), 1.0), y, Float64(-0.0625 * sin(x)))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_4 * cos(y)))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(-0.0625 * t_5), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_3)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.072], N[(N[(2.0 / N[(N[(N[Cos[y], $MachinePrecision] * t$95$4 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.245], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(-0.0625 * t$95$5), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0719999999999999946

   1. Initial program 99.1%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 55.9%

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

   if -0.0719999999999999946 < y < 0.245

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right), \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right) \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, {y}^{2}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{120} - \frac{1}{6}, y \cdot y, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-sin.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333 - 0.16666666666666666, y \cdot y, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.245 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 62.3

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

   10. Applied rewrites 62.3%

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

Alternative 9: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.057:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\cos y, \frac{t\_2}{2}, \mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right)\right) \cdot 3} + \frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot t\_3}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)} \cdot -0.020833333333333332\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_2\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.25, t\_2, \left(0.020833333333333332 \cdot \left(y \cdot y\right)\right) \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, -0.0625 \cdot t\_3, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (pow (sin y) 2.0)))
   (if (<= y -0.057)
     (+
      (/ 2.0 (* (fma (cos y) (/ t_2 2.0) (fma (cos x) (/ t_0 2.0) 1.0)) 3.0))
      (*
       (/
        (* (* (- 1.0 (cos y)) (sqrt 2.0)) t_3)
        (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
       -0.020833333333333332))
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y)))
           (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
          t_1))
        (*
         3.0
         (+
          1.0
          (fma
           0.5
           (fma (cos x) t_0 t_2)
           (*
            (* y y)
            (fma -0.25 t_2 (* (* 0.020833333333333332 (* y y)) t_2)))))))
       (/
        (fma (* (sqrt 2.0) t_1) (* -0.0625 t_3) 2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.057) {
		tmp = (2.0 / (fma(cos(y), (t_2 / 2.0), fma(cos(x), (t_0 / 2.0), 1.0)) * 3.0)) + (((((1.0 - cos(y)) * sqrt(2.0)) * t_3) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * -0.020833333333333332);
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * (1.0 + fma(0.5, fma(cos(x), t_0, t_2), ((y * y) * fma(-0.25, t_2, ((0.020833333333333332 * (y * y)) * t_2))))));
	} else {
		tmp = fma((sqrt(2.0) * t_1), (-0.0625 * t_3), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.057)
		tmp = Float64(Float64(2.0 / Float64(fma(cos(y), Float64(t_2 / 2.0), fma(cos(x), Float64(t_0 / 2.0), 1.0)) * 3.0)) + Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * t_3) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * -0.020833333333333332));
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, fma(cos(x), t_0, t_2), Float64(Float64(y * y) * fma(-0.25, t_2, Float64(Float64(0.020833333333333332 * Float64(y * y)) * t_2)))))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(-0.0625 * t_3), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.057], N[(N[(2.0 / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -0.020833333333333332), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.092], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(-0.25 * t$95$2 + N[(N[(0.020833333333333332 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0570000000000000021

   1. Initial program 99.1%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 55.9%

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

   if -0.0570000000000000021 < y < 0.091999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)}\right)} \]

       2. associate-+r+ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}\right)\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{{y}^{2}} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)}, {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.25, 3 - \sqrt{5}, \left(0.020833333333333332 \cdot \left(y \cdot y\right)\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   if 0.091999999999999998 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 62.3

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

   10. Applied rewrites 62.3%

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

Alternative 10: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.068:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_2, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_2\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.25, t\_2, \left(0.020833333333333332 \cdot \left(y \cdot y\right)\right) \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.068)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) t_2) (* 0.5 t_0)))))
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y)))
           (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
          t_1))
        (*
         3.0
         (+
          1.0
          (fma
           0.5
           (fma (cos x) t_0 t_2)
           (*
            (* y y)
            (fma -0.25 t_2 (* (* 0.020833333333333332 (* y y)) t_2)))))))
       (/
        (fma (* (sqrt 2.0) t_1) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.068) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(y) * t_2), (0.5 * t_0))));
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * (1.0 + fma(0.5, fma(cos(x), t_0, t_2), ((y * y) * fma(-0.25, t_2, ((0.020833333333333332 * (y * y)) * t_2))))));
	} else {
		tmp = fma((sqrt(2.0) * t_1), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.068)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * t_2), Float64(0.5 * t_0)))));
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, fma(cos(x), t_0, t_2), Float64(Float64(y * y) * fma(-0.25, t_2, Float64(Float64(0.020833333333333332 * Float64(y * y)) * t_2)))))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.068], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.092], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * N[(-0.25 * t$95$2 + N[(N[(0.020833333333333332 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.068000000000000005

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 23.2

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 23.0

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

   8. Applied rewrites 23.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lift--.f64 55.8

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

   11. Applied rewrites 55.8%

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

   if -0.068000000000000005 < y < 0.091999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)}\right)} \]

       2. associate-+r+ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{{y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}\right)\right)} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{{y}^{2}} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)}, {y}^{2} \cdot \left(\frac{-1}{4} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{48} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), \left(y \cdot y\right) \cdot \mathsf{fma}\left(-0.25, 3 - \sqrt{5}, \left(0.020833333333333332 \cdot \left(y \cdot y\right)\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   if 0.091999999999999998 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 62.3

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

   10. Applied rewrites 62.3%

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

Alternative 11: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.057 \lor \neg \left(y \leq 0.092\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_2}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= y -0.057) (not (<= y 0.092)))
     (/
      (fma (* (sqrt 2.0) t_2) (* -0.0625 (pow (sin y) 2.0)) 2.0)
      (fma (fma (* 0.5 (cos x)) t_1 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        t_2))
      (*
       3.0
       (+
        (+ 1.0 (* (/ t_1 2.0) (cos x)))
        (*
         (/ t_0 2.0)
         (+ 1.0 (* (* y y) (- (* 0.041666666666666664 (* y y)) 0.5))))))))))

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 tmp;
	if ((y <= -0.057) || !(y <= 0.092)) {
		tmp = fma((sqrt(2.0) * t_2), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma((0.5 * cos(x)), t_1, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_2)) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * (1.0 + ((y * y) * ((0.041666666666666664 * (y * y)) - 0.5))))));
	}
	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))
	tmp = 0.0
	if ((y <= -0.057) || !(y <= 0.092))
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_1, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5)))))));
	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]}, If[Or[LessEqual[y, -0.057], N[Not[LessEqual[y, 0.092]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0570000000000000021 or 0.091999999999999998 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 59.2

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

   10. Applied rewrites 59.2%

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

   if -0.0570000000000000021 < y < 0.091999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right)\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right)\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)\right)} \]

       7. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}\right)\right)\right)} \]

       8. lift-*.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.057 \lor \neg \left(y \leq 0.092\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.068:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(1 + \mathsf{fma}\left(0.5, \cos y \cdot t\_2, 0.5 \cdot t\_0\right)\right)}\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_2}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.068)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
      (* 3.0 (+ 1.0 (fma 0.5 (* (cos y) t_2) (* 0.5 t_0)))))
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (-
             (sin x)
             (*
              (fma
               (-
                (*
                 (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
                 (* y y))
                0.010416666666666666)
               (* y y)
               0.0625)
              y)))
           (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
          t_1))
        (*
         3.0
         (+
          (+ 1.0 (* (/ t_0 2.0) (cos x)))
          (*
           (/ t_2 2.0)
           (+ 1.0 (* (* y y) (- (* 0.041666666666666664 (* y y)) 0.5)))))))
       (/
        (fma (* (sqrt 2.0) t_1) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.068) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * (1.0 + fma(0.5, (cos(y) * t_2), (0.5 * t_0))));
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_1)) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_2 / 2.0) * (1.0 + ((y * y) * ((0.041666666666666664 * (y * y)) - 0.5))))));
	} else {
		tmp = fma((sqrt(2.0) * t_1), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.068)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_1)) / Float64(3.0 * Float64(1.0 + fma(0.5, Float64(cos(y) * t_2), Float64(0.5 * t_0)))));
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_1)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_2 / 2.0) * Float64(1.0 + Float64(Float64(y * y) * Float64(Float64(0.041666666666666664 * Float64(y * y)) - 0.5)))))));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.068], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.092], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.068000000000000005

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 23.2

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in x around 0

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

      1. lift-sin.f64 23.0

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

   8. Applied rewrites 23.0%

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

   9. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lift--.f64 55.8

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

   11. Applied rewrites 55.8%

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

   if -0.068000000000000005 < y < 0.091999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \color{blue}{{y}^{2} \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)}\right)\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right)\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot {y}^{2}} - \frac{1}{2}\right)\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot {y}^{2} - \frac{1}{2}\right)\right)\right)} \]

       7. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{1}{24} \cdot \left(y \cdot y\right) - \frac{1}{2}\right)\right)\right)} \]

       8. lift-*.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\left(1 + \left(y \cdot y\right) \cdot \left(0.041666666666666664 \cdot \left(y \cdot y\right) - 0.5\right)\right)}\right)} \]

   if 0.091999999999999998 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 62.3

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

   10. Applied rewrites 62.3%

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

Alternative 13: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot t\_2}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), t\_0, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_1, t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= y -0.0105) (not (<= y 0.028)))
     (/
      (fma (* (sqrt 2.0) t_2) (* -0.0625 (pow (sin y) 2.0)) 2.0)
      (fma (fma (* 0.5 (cos x)) t_1 1.0) 3.0 (* (* 1.5 (cos y)) t_0)))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        t_2))
      (fma
       (* -0.75 (* y y))
       t_0
       (* 3.0 (+ 1.0 (* 0.5 (fma (cos x) t_1 t_0)))))))))

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 tmp;
	if ((y <= -0.0105) || !(y <= 0.028)) {
		tmp = fma((sqrt(2.0) * t_2), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(fma((0.5 * cos(x)), t_1, 1.0), 3.0, ((1.5 * cos(y)) * t_0));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * t_2)) / fma((-0.75 * (y * y)), t_0, (3.0 * (1.0 + (0.5 * fma(cos(x), t_1, t_0)))));
	}
	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))
	tmp = 0.0
	if ((y <= -0.0105) || !(y <= 0.028))
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(fma(Float64(0.5 * cos(x)), t_1, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * t_2)) / fma(Float64(-0.75 * Float64(y * y)), t_0, Float64(3.0 * Float64(1.0 + Float64(0.5 * fma(cos(x), t_1, t_0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.0105], N[Not[LessEqual[y, 0.028]], $MachinePrecision]], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.75 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0105000000000000007 or 0.0280000000000000006 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 59.2

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

   10. Applied rewrites 59.2%

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

   if -0.0105000000000000007 < y < 0.0280000000000000006

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot {y}^{2}\right) \cdot \left(3 - \sqrt{5}\right) + \color{blue}{3} \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot {y}^{2}, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot {y}^{2}, \color{blue}{3} - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       7. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       9. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       10. distribute-lft-out N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), t\_0, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_1, t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -0.0105) (not (<= y 0.028)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (+ (fma (cos x) (/ t_1 2.0) 1.0) (* (cos y) (/ t_0 2.0)))))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        (- (cos x) (cos y))))
      (fma
       (* -0.75 (* y y))
       t_0
       (* 3.0 (+ 1.0 (* 0.5 (fma (cos x) t_1 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -0.0105) || !(y <= 0.028)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (fma(cos(x), (t_1 / 2.0), 1.0) + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * (cos(x) - cos(y)))) / fma((-0.75 * (y * y)), t_0, (3.0 * (1.0 + (0.5 * fma(cos(x), t_1, t_0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -0.0105) || !(y <= 0.028))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * Float64(cos(x) - cos(y)))) / fma(Float64(-0.75 * Float64(y * y)), t_0, Float64(3.0 * Float64(1.0 + Float64(0.5 * fma(cos(x), t_1, t_0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0105], N[Not[LessEqual[y, 0.028]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-0.75 * N[(y * y), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0105000000000000007 or 0.0280000000000000006 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 59.1%

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

   if -0.0105000000000000007 < y < 0.0280000000000000006

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot {y}^{2}\right) \cdot \left(3 - \sqrt{5}\right) + \color{blue}{3} \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot {y}^{2}, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot {y}^{2}, \color{blue}{3} - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       4. pow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       7. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       9. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

       10. distribute-lft-out N/A

           \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{-3}{4} \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   11. Applied rewrites 99.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105 \lor \neg \left(y \leq 0.028\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(-0.75 \cdot \left(y \cdot y\right), 3 - \sqrt{5}, 3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{\sqrt{5} - 1}{2}\\ \mathbf{if}\;y \leq -0.0023 \lor \neg \left(y \leq 0.0052\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, t\_1, 1\right) + \cos y \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \left(\left(1 + t\_1 \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0)) (t_1 (/ (- (sqrt 5.0) 1.0) 2.0)))
   (if (or (<= y -0.0023) (not (<= y 0.0052)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (+ (fma (cos x) t_1 1.0) (* (cos y) t_0))))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        (- (cos x) 1.0)))
      (* 3.0 (+ (+ 1.0 (* t_1 (cos x))) (* t_0 (cos y))))))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = (sqrt(5.0) - 1.0) / 2.0;
	double tmp;
	if ((y <= -0.0023) || !(y <= 0.0052)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (fma(cos(x), t_1, 1.0) + (cos(y) * t_0)));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * (cos(x) - 1.0))) / (3.0 * ((1.0 + (t_1 * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	tmp = 0.0
	if ((y <= -0.0023) || !(y <= 0.0052))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(fma(cos(x), t_1, 1.0) + Float64(cos(y) * t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * Float64(cos(x) - 1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_1 * cos(x))) + Float64(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0023], N[Not[LessEqual[y, 0.0052]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0023 or 0.0051999999999999998 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 59.1%

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

   if -0.0023 < y < 0.0051999999999999998

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \color{blue}{1}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0023 \lor \neg \left(y \leq 0.0052\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.00025 \lor \neg \left(y \leq 0.005\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, t\_1, t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= y -0.00025) (not (<= y 0.005)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (+ (fma (cos x) (/ t_1 2.0) 1.0) (* (cos y) (/ t_0 2.0)))))
     (/
      (+
       2.0
       (*
        (*
         (*
          (sqrt 2.0)
          (-
           (sin x)
           (*
            (fma
             (-
              (*
               (fma (* y y) -1.240079365079365e-5 0.0005208333333333333)
               (* y y))
              0.010416666666666666)
             (* y y)
             0.0625)
            y)))
         (fma (fma (* y y) -0.16666666666666666 1.0) y (* -0.0625 (sin x))))
        (- (cos x) (cos y))))
      (* 3.0 (+ 1.0 (* 0.5 (fma (cos x) t_1 t_0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if ((y <= -0.00025) || !(y <= 0.005)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (fma(cos(x), (t_1 / 2.0), 1.0) + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (fma(((fma((y * y), -1.240079365079365e-5, 0.0005208333333333333) * (y * y)) - 0.010416666666666666), (y * y), 0.0625) * y))) * fma(fma((y * y), -0.16666666666666666, 1.0), y, (-0.0625 * sin(x)))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + (0.5 * fma(cos(x), t_1, t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((y <= -0.00025) || !(y <= 0.005))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(fma(Float64(Float64(fma(Float64(y * y), -1.240079365079365e-5, 0.0005208333333333333) * Float64(y * y)) - 0.010416666666666666), Float64(y * y), 0.0625) * y))) * fma(fma(Float64(y * y), -0.16666666666666666, 1.0), y, Float64(-0.0625 * sin(x)))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(0.5 * fma(cos(x), t_1, t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.00025], N[Not[LessEqual[y, 0.005]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -1.240079365079365e-5 + 0.0005208333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.0625), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000001e-4 or 0.0050000000000000001 < y

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 59.1%

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

   if -2.5000000000000001e-4 < y < 0.0050000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) - \frac{1}{16} \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right) \cdot \sin x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + \frac{-1}{16} \cdot \sin \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(1 + \frac{-1}{6} \cdot {y}^{2}, \color{blue}{y}, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot {y}^{2} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left({y}^{2} \cdot \frac{-1}{6} + 1, y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. pow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lift-sin.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

       2. distribute-lft-out N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right)\right)} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right)\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)} \]

       7. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - \color{blue}{1}, 3 - \sqrt{5}\right)\right)} \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{80640}, \frac{1}{1920}\right) \cdot \left(y \cdot y\right) - \frac{1}{96}, y \cdot y, \frac{1}{16}\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{-1}{6}, 1\right), y, \frac{-1}{16} \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)} \]

       9. lift--.f64 98.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)} \]

   11. Applied rewrites 98.4%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00025 \lor \neg \left(y \leq 0.005\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -1.240079365079365 \cdot 10^{-5}, 0.0005208333333333333\right) \cdot \left(y \cdot y\right) - 0.010416666666666666, y \cdot y, 0.0625\right) \cdot y\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right), y, -0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right) + \cos y \cdot \frac{t\_1}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0011) (not (<= x 13.0)))
     (*
      (/
       (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (- (cos x) 1.0))))
       (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))
      0.3333333333333333)
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (+ (fma (cos x) (/ t_0 2.0) 1.0) (* (cos y) (/ t_1 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0011) || !(x <= 13.0)) {
		tmp = ((2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (fma(cos(x), (t_0 / 2.0), 1.0) + (cos(y) * (t_1 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0011) || !(x <= 13.0))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(fma(cos(x), Float64(t_0 / 2.0), 1.0) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0011], N[Not[LessEqual[x, 13.0]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00110000000000000007 or 13 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -0.00110000000000000007 < x < 13

   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. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sqrt.f64 98.3

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

   7. Applied rewrites 98.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

   9. Applied rewrites 98.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, t\_0, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0011) (not (<= x 13.0)))
     (*
      (/
       (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (- (cos x) 1.0))))
       (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))
      0.3333333333333333)
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y)))))
      (fma (fma (* 0.5 (cos x)) t_0 1.0) 3.0 (* (* 1.5 (cos y)) t_1))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0011) || !(x <= 13.0)) {
		tmp = ((2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(y), 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / fma(fma((0.5 * cos(x)), t_0, 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0011) || !(x <= 13.0))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))) / fma(fma(Float64(0.5 * cos(x)), t_0, 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0011], N[Not[LessEqual[x, 13.0]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(2.0 + 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]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00110000000000000007 or 13 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -0.00110000000000000007 < x < 13

   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. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 98.3

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

   10. Applied rewrites 98.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(0.5, t\_0, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -4.1e-5) (not (<= x 1.7e-8)))
     (*
      (/
       (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (- (cos x) 1.0))))
       (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))
      0.3333333333333333)
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (* 1.5 (cos y)) t_1 (* (fma 0.5 t_0 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -4.1e-5) || !(x <= 1.7e-8)) {
		tmp = ((2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_1, (fma(0.5, t_0, 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.1e-5) || !(x <= 1.7e-8))
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_1, Float64(fma(0.5, t_0, 1.0) * 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-5], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000005e-5 or 1.7e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 56.9

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

   8. Applied rewrites 56.9%

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

   if -4.10000000000000005e-5 < x < 1.7e-8

   1. Initial program 99.7%

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

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(0.5, t\_0, 1\right) \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -4.1e-5) (not (<= x 1.7e-8)))
     (/
      (*
       (fma (* (- (cos x) 1.0) (sqrt 2.0)) (* (pow (sin x) 2.0) -0.0625) 2.0)
       0.3333333333333333)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma (* 1.5 (cos y)) t_1 (* (fma 0.5 t_0 1.0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -4.1e-5) || !(x <= 1.7e-8)) {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_1, (fma(0.5, t_0, 1.0) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.1e-5) || !(x <= 1.7e-8))
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_1, Float64(fma(0.5, t_0, 1.0) * 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-5], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000005e-5 or 1.7e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.7%

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

   if -4.10000000000000005e-5 < x < 1.7e-8

   1. Initial program 99.7%

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

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{1 + 0.5 \cdot \mathsf{fma}\left(\cos y, t\_1, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -4.1e-5) (not (<= x 1.7e-8)))
     (/
      (*
       (fma (* (- (cos x) 1.0) (sqrt 2.0)) (* (pow (sin x) 2.0) -0.0625) 2.0)
       0.3333333333333333)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y)))))
       (+ 1.0 (* 0.5 (fma (cos y) t_1 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -4.1e-5) || !(x <= 1.7e-8)) {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((-0.0625 * pow(sin(y), 2.0)) * (sqrt(2.0) * (1.0 - cos(y))))) / (1.0 + (0.5 * fma(cos(y), t_1, t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.1e-5) || !(x <= 1.7e-8))
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))) / Float64(1.0 + Float64(0.5 * fma(cos(y), t_1, t_0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-5], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + 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]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000005e-5 or 1.7e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.7%

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

   if -4.10000000000000005e-5 < x < 1.7e-8

   1. Initial program 99.7%

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

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   10. Applied rewrites 98.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, t\_1\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -4.1e-5) (not (<= x 1.7e-8)))
     (/
      (*
       (fma (* (- (cos x) 1.0) (sqrt 2.0)) (* (pow (sin x) 2.0) -0.0625) 2.0)
       0.3333333333333333)
      (fma (fma t_0 (cos x) t_1) 0.5 1.0))
     (*
      (/
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -4.1e-5) || !(x <= 1.7e-8)) {
		tmp = (fma(((cos(x) - 1.0) * sqrt(2.0)), (pow(sin(x), 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0);
	} else {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.1e-5) || !(x <= 1.7e-8))
		tmp = Float64(Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64((sin(x) ^ 2.0) * -0.0625), 2.0) * 0.3333333333333333) / fma(fma(t_0, cos(x), t_1), 0.5, 1.0));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-5], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000005e-5 or 1.7e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.7%

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

   if -4.10000000000000005e-5 < x < 1.7e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 98.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, {\sin x}^{2} \cdot -0.0625, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -4.1e-5) (not (<= x 1.7e-8)))
     (*
      (/
       (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
       (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
      0.3333333333333333)
     (*
      (/
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
       (fma 0.5 (fma t_1 (cos y) t_0) 1.0))
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -4.1e-5) || !(x <= 1.7e-8)) {
		tmp = (fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -4.1e-5) || !(x <= 1.7e-8))
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_1, cos(y), t_0), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.1e-5], N[Not[LessEqual[x, 1.7e-8]], $MachinePrecision]], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $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$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.10000000000000005e-5 or 1.7e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 55.7%

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

   if -4.10000000000000005e-5 < x < 1.7e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

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

   5. Applied rewrites 98.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 59.5%

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

Alternative 25: 60.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 59.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. associate-+r- N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. lower--.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lift-cos.f64 59.5

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

7. Applied rewrites 59.5%

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

Alternative 26: 45.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(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. Add Preprocessing
3. 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

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift--.f64 63.0

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

10. Applied rewrites 63.0%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 46.7%

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

Alternative 27: 43.5% accurate, 6.1× 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. Add Preprocessing

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 59.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.5%

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

Alternative 28: 41.0% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

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. Add Preprocessing

3. Taylor expanded in y around 0

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

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

5. Applied rewrites 59.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
7. Step-by-step derivation

8. Applied rewrites 42.4%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025060 
(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))))))