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

Percentage Accurate: 99.2% → 99.3%

Time: 8.9s

Alternatives: 26

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: 99.2% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)\\ \mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{t\_0}, 0.3333333333333333, \frac{0.6666666666666666}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
          1.0)))
   (fma
    (/
     (*
      (* (sqrt 2.0) (- (cos x) (cos y)))
      (* (- (sin y) (* (sin x) 0.0625)) (- (sin x) (* (sin y) 0.0625))))
     t_0)
    0.3333333333333333
    (/ 0.6666666666666666 t_0))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0);
	return fma((((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625)))) / t_0), 0.3333333333333333, (0.6666666666666666 / t_0));
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)
	return fma(Float64(Float64(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625)))) / t_0), 0.3333333333333333, Float64(0.6666666666666666 / t_0))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, 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[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333 + N[(0.6666666666666666 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.2%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right)} \]
7. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return fma((((sin(y) - (sin(x) * 0.0625)) * (sin(x) - (sin(y) * 0.0625))) * (cos(x) - cos(y))), sqrt(2.0), 2.0) / fma(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(Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(sin(x) - Float64(sin(y) * 0.0625))) * Float64(cos(x) - cos(y))), sqrt(2.0), 2.0) / fma(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[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(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 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (* 0.0625 (sin x))))
    (- (cos x) (cos y))))
  (*
   3.0
   (fma
    0.5
    (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
    1.0))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * sin(x)))) * (cos(x) - cos(y)))) / (3.0 * fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * sin(x)))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(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[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sin.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 99.2%

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

Alternative 6: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := 2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_2\\ t_4 := \sqrt{5} - 1\\ t_5 := \frac{t\_4}{2}\\ t_6 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_5, 1\right), 3, \left(\cos y \cdot t\_1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_6}{16}\right)\right) \cdot \left(t\_6 - \frac{\sin x}{16}\right)\right) \cdot t\_2}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_4, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{3 \cdot \left(\left(1 + t\_5 \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3
         (+
          2.0
          (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_2)))
        (t_4 (- (sqrt 5.0) 1.0))
        (t_5 (/ t_4 2.0))
        (t_6 (* y (- 1.0 (* 0.16666666666666666 (* y y))))))
   (if (<= y -0.205)
     (/ t_3 (fma (fma (cos x) t_5 1.0) 3.0 (* (* (cos y) t_1) 3.0)))
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_6 16.0))) (- t_6 (/ (sin x) 16.0)))
          t_2))
        (* 3.0 (fma 0.5 (fma t_4 (cos x) (* t_0 (cos y))) 1.0)))
       (/ t_3 (* 3.0 (+ (+ 1.0 (* t_5 (cos x))) (* t_1 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = 2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_2);
	double t_4 = sqrt(5.0) - 1.0;
	double t_5 = t_4 / 2.0;
	double t_6 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double tmp;
	if (y <= -0.205) {
		tmp = t_3 / fma(fma(cos(x), t_5, 1.0), 3.0, ((cos(y) * t_1) * 3.0));
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_6 / 16.0))) * (t_6 - (sin(x) / 16.0))) * t_2)) / (3.0 * fma(0.5, fma(t_4, cos(x), (t_0 * cos(y))), 1.0));
	} else {
		tmp = t_3 / (3.0 * ((1.0 + (t_5 * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_2))
	t_4 = Float64(sqrt(5.0) - 1.0)
	t_5 = Float64(t_4 / 2.0)
	t_6 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	tmp = 0.0
	if (y <= -0.205)
		tmp = Float64(t_3 / fma(fma(cos(x), t_5, 1.0), 3.0, Float64(Float64(cos(y) * t_1) * 3.0)));
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_6 / 16.0))) * Float64(t_6 - Float64(sin(x) / 16.0))) * t_2)) / Float64(3.0 * fma(0.5, fma(t_4, cos(x), Float64(t_0 * cos(y))), 1.0)));
	else
		tmp = Float64(t_3 / Float64(3.0 * Float64(Float64(1.0 + Float64(t_5 * cos(x))) + Float64(t_1 * 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[(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[(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$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / 2.0), $MachinePrecision]}, Block[{t$95$6 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.205], N[(t$95$3 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$5 + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] * 3.0), $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[(t$95$6 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$6 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$4 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(3.0 * N[(N[(1.0 + N[(t$95$5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.204999999999999988

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-sin.f64 64.8

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

   6. Applied rewrites 64.8%

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

   if -0.204999999999999988 < y < 0.091999999999999998

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 0.091999999999999998 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.0

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

   4. Applied rewrites 63.0%

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

Alternative 7: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_1}{3 \cdot \left(\left(1 + \frac{t\_2}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ t_4 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_4}{16}\right)\right) \cdot \left(t\_4 - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_1))
          (* 3.0 (+ (+ 1.0 (* (/ t_2 2.0) (cos x))) (* (/ t_0 2.0) (cos y))))))
        (t_4 (* y (- 1.0 (* 0.16666666666666666 (* y y))))))
   (if (<= y -0.205)
     t_3
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_4 16.0))) (- t_4 (/ (sin x) 16.0)))
          t_1))
        (* 3.0 (fma 0.5 (fma t_2 (cos x) (* t_0 (cos y))) 1.0)))
       t_3))))

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 t_3 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_1)) / (3.0 * ((1.0 + ((t_2 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double t_4 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double tmp;
	if (y <= -0.205) {
		tmp = t_3;
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_4 / 16.0))) * (t_4 - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(0.5, fma(t_2, cos(x), (t_0 * cos(y))), 1.0));
	} else {
		tmp = t_3;
	}
	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)
	t_3 = 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(Float64(1.0 + Float64(Float64(t_2 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	t_4 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	tmp = 0.0
	if (y <= -0.205)
		tmp = t_3;
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_4 / 16.0))) * Float64(t_4 - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * fma(0.5, fma(t_2, cos(x), Float64(t_0 * cos(y))), 1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = 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[(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]}, Block[{t$95$4 = N[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.205], t$95$3, If[LessEqual[y, 0.092], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(t$95$4 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.204999999999999988 or 0.091999999999999998 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.9

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

   4. Applied rewrites 63.9%

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

   if -0.204999999999999988 < y < 0.091999999999999998

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

Alternative 8: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := y \cdot \left(1 - 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ t_2 := 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{t\_2}\\ \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{t\_1}{16}\right)\right) \cdot \left(t\_1 - \frac{\sin x}{16}\right)\right) \cdot t\_0}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* y (- 1.0 (* 0.16666666666666666 (* y y)))))
        (t_2
         (*
          3.0
          (fma
           0.5
           (fma (- (sqrt 5.0) 1.0) (cos x) (* (- 3.0 (sqrt 5.0)) (cos y)))
           1.0)))
        (t_3
         (/
          (+
           2.0
           (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
          t_2)))
   (if (<= y -0.205)
     t_3
     (if (<= y 0.092)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ t_1 16.0))) (- t_1 (/ (sin x) 16.0)))
          t_0))
        t_2)
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = y * (1.0 - (0.16666666666666666 * (y * y)));
	double t_2 = 3.0 * fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0);
	double t_3 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / t_2;
	double tmp;
	if (y <= -0.205) {
		tmp = t_3;
	} else if (y <= 0.092) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (t_1 / 16.0))) * (t_1 - (sin(x) / 16.0))) * t_0)) / t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(y * Float64(1.0 - Float64(0.16666666666666666 * Float64(y * y))))
	t_2 = Float64(3.0 * fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0))
	t_3 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / t_2)
	tmp = 0.0
	if (y <= -0.205)
		tmp = t_3;
	elseif (y <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(t_1 / 16.0))) * Float64(t_1 - Float64(sin(x) / 16.0))) * t_0)) / t_2);
	else
		tmp = t_3;
	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[(y * N[(1.0 - N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[y, -0.205], t$95$3, If[LessEqual[y, 0.092], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(t$95$1 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.204999999999999988 or 0.091999999999999998 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.9

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

   7. Applied rewrites 63.9%

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

   if -0.204999999999999988 < y < 0.091999999999999998

   1. Initial program 99.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

Alternative 9: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - 0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)\\ t_3 := \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot t\_1\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{t\_2}\right)\\ \mathbf{if}\;x \leq -0.105:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.28:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(t\_0 - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{t\_0}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* 0.16666666666666666 (* x x)))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (fma 0.5 (fma t_1 (cos x) (* (- 3.0 (sqrt 5.0)) (cos y))) 1.0))
        (t_3
         (fma
          (/
           (*
            (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
            (* (sqrt 2.0) (- (cos x) 1.0)))
           (- 1.0 (* -0.5 (- (+ 3.0 (* (cos x) t_1)) (sqrt 5.0)))))
          0.3333333333333333
          (/ 0.6666666666666666 t_2))))
   (if (<= x -0.105)
     t_3
     (if (<= x 0.28)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- t_0 (/ (sin y) 16.0))) (- (sin y) (/ t_0 16.0)))
          (- (cos x) (cos y))))
        (* 3.0 t_2))
       t_3))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - (0.16666666666666666 * (x * x)));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(0.5, fma(t_1, cos(x), ((3.0 - sqrt(5.0)) * cos(y))), 1.0);
	double t_3 = fma((((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 - (-0.5 * ((3.0 + (cos(x) * t_1)) - sqrt(5.0))))), 0.3333333333333333, (0.6666666666666666 / t_2));
	double tmp;
	if (x <= -0.105) {
		tmp = t_3;
	} else if (x <= 0.28) {
		tmp = (2.0 + (((sqrt(2.0) * (t_0 - (sin(y) / 16.0))) * (sin(y) - (t_0 / 16.0))) * (cos(x) - cos(y)))) / (3.0 * t_2);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = fma(0.5, fma(t_1, cos(x), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 1.0)
	t_3 = fma(Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 - Float64(-0.5 * Float64(Float64(3.0 + Float64(cos(x) * t_1)) - sqrt(5.0))))), 0.3333333333333333, Float64(0.6666666666666666 / t_2))
	tmp = 0.0
	if (x <= -0.105)
		tmp = t_3;
	elseif (x <= 0.28)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(t_0 - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(t_0 / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * t_2));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(0.6666666666666666 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.105], t$95$3, If[LessEqual[x, 0.28], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(t$95$0 / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * t$95$2), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.104999999999999996 or 0.28000000000000003 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right)} \]

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right) \]

   if -0.104999999999999996 < x < 0.28000000000000003

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

Alternative 10: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot t\_0\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)}\right)\\ \mathbf{if}\;x \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.084:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{t\_1}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (/
           (*
            (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
            (* (sqrt 2.0) (- (cos x) 1.0)))
           (- 1.0 (* -0.5 (- (+ 3.0 (* (cos x) t_0)) (sqrt 5.0)))))
          0.3333333333333333
          (/
           0.6666666666666666
           (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0)))))
   (if (<= x -0.05)
     t_2
     (if (<= x 0.084)
       (/
        (+
         2.0
         (*
          (*
           (*
            (sqrt 2.0)
            (fma
             x
             (- 1.0 (* 0.16666666666666666 (* x x)))
             (* -0.0625 (sin y))))
           (- (sin y) (/ x 16.0)))
          (- (cos x) (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_0 2.0) (cos x))) (* (/ t_1 2.0) (cos y)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 - (-0.5 * ((3.0 + (cos(x) * t_0)) - sqrt(5.0))))), 0.3333333333333333, (0.6666666666666666 / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)));
	double tmp;
	if (x <= -0.05) {
		tmp = t_2;
	} else if (x <= 0.084) {
		tmp = (2.0 + (((sqrt(2.0) * fma(x, (1.0 - (0.16666666666666666 * (x * x))), (-0.0625 * sin(y)))) * (sin(y) - (x / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + ((t_1 / 2.0) * cos(y))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 - Float64(-0.5 * Float64(Float64(3.0 + Float64(cos(x) * t_0)) - sqrt(5.0))))), 0.3333333333333333, Float64(0.6666666666666666 / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)))
	tmp = 0.0
	if (x <= -0.05)
		tmp = t_2;
	elseif (x <= 0.084)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(x, Float64(1.0 - Float64(0.16666666666666666 * Float64(x * x))), Float64(-0.0625 * sin(y)))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(t_1 / 2.0) * cos(y)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(0.6666666666666666 / 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]), $MachinePrecision]}, If[LessEqual[x, -0.05], t$95$2, If[LessEqual[x, 0.084], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(1.0 - N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 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[(t$95$0 / 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], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.050000000000000003 or 0.0840000000000000052 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right)} \]

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right) \]

   if -0.050000000000000003 < x < 0.0840000000000000052

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-sin.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(x, 1 - 0.16666666666666666 \cdot \left(x \cdot x\right), -0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot t\_0\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)}\right)\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.025:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right), 3, \left(\cos y \cdot \frac{t\_1}{2}\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (/
           (*
            (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
            (* (sqrt 2.0) (- (cos x) 1.0)))
           (- 1.0 (* -0.5 (- (+ 3.0 (* (cos x) t_0)) (sqrt 5.0)))))
          0.3333333333333333
          (/
           0.6666666666666666
           (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0)))))
   (if (<= x -0.032)
     t_2
     (if (<= x 0.025)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- (cos x) (cos y))))
        (fma
         (fma (cos x) (/ t_0 2.0) 1.0)
         3.0
         (* (* (cos y) (/ t_1 2.0)) 3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 - (-0.5 * ((3.0 + (cos(x) * t_0)) - sqrt(5.0))))), 0.3333333333333333, (0.6666666666666666 / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)));
	double tmp;
	if (x <= -0.032) {
		tmp = t_2;
	} else if (x <= 0.025) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (cos(x) - cos(y)))) / fma(fma(cos(x), (t_0 / 2.0), 1.0), 3.0, ((cos(y) * (t_1 / 2.0)) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 - Float64(-0.5 * Float64(Float64(3.0 + Float64(cos(x) * t_0)) - sqrt(5.0))))), 0.3333333333333333, Float64(0.6666666666666666 / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)))
	tmp = 0.0
	if (x <= -0.032)
		tmp = t_2;
	elseif (x <= 0.025)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(cos(x) - cos(y)))) / fma(fma(cos(x), Float64(t_0 / 2.0), 1.0), 3.0, Float64(Float64(cos(y) * Float64(t_1 / 2.0)) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(0.6666666666666666 / 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]), $MachinePrecision]}, If[LessEqual[x, -0.032], t$95$2, If[LessEqual[x, 0.025], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.032000000000000001 or 0.025000000000000001 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right)} \]

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right) \]

   if -0.032000000000000001 < x < 0.025000000000000001

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.5%

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

Alternative 12: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left(x \cdot x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot t\_1\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2 \cdot \cos y\right), 1\right)}\right)\\ \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_0 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot t\_0\right) + \frac{t\_2}{2} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* x x))))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (fma
          (/
           (*
            (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x)))))
            (* (sqrt 2.0) (- (cos x) 1.0)))
           (- 1.0 (* -0.5 (- (+ 3.0 (* (cos x) t_1)) (sqrt 5.0)))))
          0.3333333333333333
          (/
           0.6666666666666666
           (fma 0.5 (fma t_1 (cos x) (* t_2 (cos y))) 1.0)))))
   (if (<= x -0.0045)
     t_3
     (if (<= x 0.0055)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_0 (cos y))))
        (* 3.0 (+ (+ 1.0 (* (/ t_1 2.0) t_0)) (* (/ t_2 2.0) (cos y)))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (0.5 * (x * x));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma((((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * (sqrt(2.0) * (cos(x) - 1.0))) / (1.0 - (-0.5 * ((3.0 + (cos(x) * t_1)) - sqrt(5.0))))), 0.3333333333333333, (0.6666666666666666 / fma(0.5, fma(t_1, cos(x), (t_2 * cos(y))), 1.0)));
	double tmp;
	if (x <= -0.0045) {
		tmp = t_3;
	} else if (x <= 0.0055) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_0 - cos(y)))) / (3.0 * ((1.0 + ((t_1 / 2.0) * t_0)) + ((t_2 / 2.0) * cos(y))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(0.5 * Float64(x * x)))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))) / Float64(1.0 - Float64(-0.5 * Float64(Float64(3.0 + Float64(cos(x) * t_1)) - sqrt(5.0))))), 0.3333333333333333, Float64(0.6666666666666666 / fma(0.5, fma(t_1, cos(x), Float64(t_2 * cos(y))), 1.0)))
	tmp = 0.0
	if (x <= -0.0045)
		tmp = t_3;
	elseif (x <= 0.0055)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_0 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * t_0)) + Float64(Float64(t_2 / 2.0) * cos(y)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(-0.5 * N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(0.6666666666666666 / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0045], t$95$3, If[LessEqual[x, 0.0055], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$1 / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00449999999999999966 or 0.0054999999999999997 < x

   1. Initial program 98.9%

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \sin x \cdot 0.0625\right) \cdot \left(\sin x - \sin y \cdot 0.0625\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right)} \]

   7. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 59.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}{1 - -0.5 \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)}, 0.3333333333333333, \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}\right) \]

   if -0.00449999999999999966 < x < 0.0054999999999999997

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 99.5

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

   13. Applied rewrites 99.5%

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

Alternative 13: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2} \cdot \cos y\\ t_1 := 1 - 0.5 \cdot \left(x \cdot x\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \frac{2 + \left(\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))
        (t_1 (- 1.0 (* 0.5 (* x x))))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3
         (/
          (+
           2.0
           (*
            (* (* -0.0625 (- 0.5 (* 0.5 (cos (+ x x))))) (sqrt 2.0))
            (- (cos x) (cos y))))
          (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) t_0)))))
   (if (<= x -0.0045)
     t_3
     (if (<= x 0.0055)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) t_0)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = ((3.0 - sqrt(5.0)) / 2.0) * cos(y);
	double t_1 = 1.0 - (0.5 * (x * x));
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_2 * cos(x))) + t_0));
	double tmp;
	if (x <= -0.0045) {
		tmp = t_3;
	} else if (x <= 0.0055) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y)
    t_1 = 1.0d0 - (0.5d0 * (x * x))
    t_2 = (sqrt(5.0d0) - 1.0d0) / 2.0d0
    t_3 = (2.0d0 + ((((-0.0625d0) * (0.5d0 - (0.5d0 * cos((x + x))))) * sqrt(2.0d0)) * (cos(x) - cos(y)))) / (3.0d0 * ((1.0d0 + (t_2 * cos(x))) + t_0))
    if (x <= (-0.0045d0)) then
        tmp = t_3
    else if (x <= 0.0055d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (sin(y) / 16.0d0))) * (sin(y) - (x / 16.0d0))) * (t_1 - cos(y)))) / (3.0d0 * ((1.0d0 + (t_2 * t_1)) + t_0))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y);
	double t_1 = 1.0 - (0.5 * (x * x));
	double t_2 = (Math.sqrt(5.0) - 1.0) / 2.0;
	double t_3 = (2.0 + (((-0.0625 * (0.5 - (0.5 * Math.cos((x + x))))) * Math.sqrt(2.0)) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((1.0 + (t_2 * Math.cos(x))) + t_0));
	double tmp;
	if (x <= -0.0045) {
		tmp = t_3;
	} else if (x <= 0.0055) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (Math.sin(y) / 16.0))) * (Math.sin(y) - (x / 16.0))) * (t_1 - Math.cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y)
	t_1 = 1.0 - (0.5 * (x * x))
	t_2 = (math.sqrt(5.0) - 1.0) / 2.0
	t_3 = (2.0 + (((-0.0625 * (0.5 - (0.5 * math.cos((x + x))))) * math.sqrt(2.0)) * (math.cos(x) - math.cos(y)))) / (3.0 * ((1.0 + (t_2 * math.cos(x))) + t_0))
	tmp = 0
	if x <= -0.0045:
		tmp = t_3
	elif x <= 0.0055:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (math.sin(y) / 16.0))) * (math.sin(y) - (x / 16.0))) * (t_1 - math.cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + t_0))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(x * x)))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + t_0)))
	tmp = 0.0
	if (x <= -0.0045)
		tmp = t_3;
	elseif (x <= 0.0055)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((3.0 - sqrt(5.0)) / 2.0) * cos(y);
	t_1 = 1.0 - (0.5 * (x * x));
	t_2 = (sqrt(5.0) - 1.0) / 2.0;
	t_3 = (2.0 + (((-0.0625 * (0.5 - (0.5 * cos((x + x))))) * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (t_2 * cos(x))) + t_0));
	tmp = 0.0;
	if (x <= -0.0045)
		tmp = t_3;
	elseif (x <= 0.0055)
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + t_0));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0045], t$95$3, If[LessEqual[x, 0.0055], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00449999999999999966 or 0.0054999999999999997 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 3.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

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

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-sqrt.f64 59.4

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

   10. Applied rewrites 59.4%

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

   if -0.00449999999999999966 < x < 0.0054999999999999997

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 99.5

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

   13. Applied rewrites 99.5%

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

Alternative 14: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := 1 - 0.5 \cdot \left(x \cdot x\right)\\ t_2 := \frac{\sqrt{5} - 1}{2}\\ t_3 := \frac{3 - \sqrt{5}}{2} \cdot \cos y\\ t_4 := 3 \cdot \left(\left(1 + t\_2 \cdot \cos x\right) + t\_3\right)\\ \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot t\_0}{t\_4}\\ \mathbf{elif}\;x \leq 0.0055:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{x}{16}\right)\right) \cdot \left(t\_1 - \cos y\right)}{3 \cdot \left(\left(1 + t\_2 \cdot t\_1\right) + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_1 (- 1.0 (* 0.5 (* x x))))
        (t_2 (/ (- (sqrt 5.0) 1.0) 2.0))
        (t_3 (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))
        (t_4 (* 3.0 (+ (+ 1.0 (* t_2 (cos x))) t_3))))
   (if (<= x -0.0045)
     (/ (+ 2.0 (* (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_0)) t_4)
     (if (<= x 0.0055)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- x (/ (sin y) 16.0))) (- (sin y) (/ x 16.0)))
          (- t_1 (cos y))))
        (* 3.0 (+ (+ 1.0 (* t_2 t_1)) t_3)))
       (/ (fma t_0 (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625) 2.0) t_4)))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = 1.0 - (0.5 * (x * x));
	double t_2 = (sqrt(5.0) - 1.0) / 2.0;
	double t_3 = ((3.0 - sqrt(5.0)) / 2.0) * cos(y);
	double t_4 = 3.0 * ((1.0 + (t_2 * cos(x))) + t_3);
	double tmp;
	if (x <= -0.0045) {
		tmp = (2.0 + ((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))) * t_0)) / t_4;
	} else if (x <= 0.0055) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (sin(y) / 16.0))) * (sin(y) - (x / 16.0))) * (t_1 - cos(y)))) / (3.0 * ((1.0 + (t_2 * t_1)) + t_3));
	} else {
		tmp = fma(t_0, ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) / t_4;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(1.0 - Float64(0.5 * Float64(x * x)))
	t_2 = Float64(Float64(sqrt(5.0) - 1.0) / 2.0)
	t_3 = Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y))
	t_4 = Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * cos(x))) + t_3))
	tmp = 0.0
	if (x <= -0.0045)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) * t_0)) / t_4);
	elseif (x <= 0.0055)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(x / 16.0))) * Float64(t_1 - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(t_2 * t_1)) + t_3)));
	else
		tmp = Float64(fma(t_0, Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) / t_4);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 * N[(N[(1.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0045], N[(N[(2.0 + N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[x, 0.0055], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(x / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00449999999999999966

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-sqrt.f64 60.4

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

   4. Applied rewrites 60.4%

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

   if -0.00449999999999999966 < x < 0.0054999999999999997

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in x around 0

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

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

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

       2. lower--.f64 N/A

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

       3. metadata-eval N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 99.5

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

   13. Applied rewrites 99.5%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 58.3%

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

Alternative 15: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := \mathsf{fma}\left(y \cdot y, -0.5, 1\right)\\ t_2 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_3 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_2 + t\_0 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.0125:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 0.0072:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - t\_1\right)}{3 \cdot \left(t\_2 + t\_0 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (fma (* y y) -0.5 1.0))
        (t_2 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_3
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (* 3.0 (+ t_2 (* t_0 (cos y)))))))
   (if (<= y -0.0125)
     t_3
     (if (<= y 0.0072)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) t_1)))
        (* 3.0 (+ t_2 (* t_0 t_1))))
       t_3))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = fma((y * y), -0.5, 1.0);
	double t_2 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_3 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_2 + (t_0 * cos(y))));
	double tmp;
	if (y <= -0.0125) {
		tmp = t_3;
	} else if (y <= 0.0072) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - t_1))) / (3.0 * (t_2 + (t_0 * t_1)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = fma(Float64(y * y), -0.5, 1.0)
	t_2 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_3 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_2 + Float64(t_0 * cos(y)))))
	tmp = 0.0
	if (y <= -0.0125)
		tmp = t_3;
	elseif (y <= 0.0072)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - t_1))) / Float64(3.0 * Float64(t_2 + Float64(t_0 * t_1))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0125], t$95$3, If[LessEqual[y, 0.0072], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.012500000000000001 or 0.0071999999999999998 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.2%

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

   if -0.012500000000000001 < y < 0.0071999999999999998

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 99.4

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

   13. Applied rewrites 99.4%

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

Alternative 16: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{3 - \sqrt{5}}{2}\\ t_1 := 1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\\ t_2 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(t\_1 + t\_0 \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -0.0007:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.00255:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{y}{16}\right)\right) \cdot \left(y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{3 \cdot \left(t\_1 + t\_0 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_1 (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))))
        (t_2
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (* 3.0 (+ t_1 (* t_0 (cos y)))))))
   (if (<= y -0.0007)
     t_2
     (if (<= y 0.00255)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (- (sin x) (/ y 16.0))) (- y (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (* 3.0 (+ t_1 (* t_0 1.0))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = (3.0 - sqrt(5.0)) / 2.0;
	double t_1 = 1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x));
	double t_2 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (t_1 + (t_0 * cos(y))));
	double tmp;
	if (y <= -0.0007) {
		tmp = t_2;
	} else if (y <= 0.00255) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (y / 16.0))) * (y - (sin(x) / 16.0))) * (cos(x) - 1.0))) / (3.0 * (t_1 + (t_0 * 1.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_1 = Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x)))
	t_2 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(t_1 + Float64(t_0 * cos(y)))))
	tmp = 0.0
	if (y <= -0.0007)
		tmp = t_2;
	elseif (y <= 0.00255)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(y / 16.0))) * Float64(y - Float64(sin(x) / 16.0))) * Float64(cos(x) - 1.0))) / Float64(3.0 * Float64(t_1 + Float64(t_0 * 1.0))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0007], t$95$2, If[LessEqual[y, 0.00255], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(y / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(t$95$0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999993e-4 or 0.0025500000000000002 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

   if -6.99999999999999993e-4 < y < 0.0025500000000000002

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 99.1%

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

Alternative 17: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_0}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00088:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, \frac{4}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (fma
           (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* (/ t_0 2.0) (cos x)))
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)))))))
   (if (<= y -4.3e-6)
     t_1
     (if (<= y 0.00088)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_0 (cos x) (/ 4.0 (+ 3.0 (sqrt 5.0)))) 1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * ((1.0 + ((t_0 / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	double tmp;
	if (y <= -4.3e-6) {
		tmp = t_1;
	} else if (y <= 0.00088) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), (4.0 / (3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_0 / 2.0) * cos(x))) + Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)))))
	tmp = 0.0
	if (y <= -4.3e-6)
		tmp = t_1;
	elseif (y <= 0.00088)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(4.0 / Float64(3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$0 / 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]}, If[LessEqual[y, -4.3e-6], t$95$1, If[LessEqual[y, 0.00088], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.30000000000000033e-6 or 8.80000000000000031e-4 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 60.3%

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

   if -4.30000000000000033e-6 < y < 8.80000000000000031e-4

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-+.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{9 \cdot 9 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {\left(\sqrt{5}\right)}^{2} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      8. sqrt-pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{1} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      14. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {\left(\sqrt{5}\right)}^{2}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      15. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{\left(\frac{2}{2}\right)}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{1}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot 5}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 25}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      23. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {\left(\sqrt{5}\right)}^{2}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      24. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{\left(\frac{2}{2}\right)}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{1}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

   8. Applied rewrites 98.9%

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

Alternative 18: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\left(\cos x - 1\right) \cdot \sqrt{2}, \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right) \cdot -0.0625, 2\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(0.5, t\_1, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (- (cos x) 1.0) (sqrt 2.0))
           (* (- 0.5 (* (cos (+ x x)) 0.5)) -0.0625)
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))
   (if (<= x -1.5e-5)
     t_2
     (if (<= x 1.38e-14)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* (fma 0.5 t_1 1.0) 3.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma(((cos(x) - 1.0) * sqrt(2.0)), ((0.5 - (cos((x + x)) * 0.5)) * -0.0625), 2.0) / (3.0 * ((1.0 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	double tmp;
	if (x <= -1.5e-5) {
		tmp = t_2;
	} else if (x <= 1.38e-14) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_0, (fma(0.5, t_1, 1.0) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) * -0.0625), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))))
	tmp = 0.0
	if (x <= -1.5e-5)
		tmp = t_2;
	elseif (x <= 1.38e-14)
		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_0, Float64(fma(0.5, t_1, 1.0) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(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]}, If[LessEqual[x, -1.5e-5], t$95$2, If[LessEqual[x, 1.38e-14], 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$0 + N[(N[(0.5 * t$95$1 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000004e-5 or 1.38000000000000002e-14 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 59.6%

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

   if -1.50000000000000004e-5 < x < 1.38000000000000002e-14

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.3%

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

Alternative 19: 78.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, \frac{4}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(0.5, t\_0, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (fma t_0 (cos x) (/ 4.0 (+ 3.0 (sqrt 5.0)))) 1.0))
          0.3333333333333333)))
   (if (<= x -1.5e-5)
     t_1
     (if (<= x 1.38e-14)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) (- 3.0 (sqrt 5.0)) (* (fma 0.5 t_0 1.0) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), (4.0 / (3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -1.5e-5) {
		tmp = t_1;
	} else if (x <= 1.38e-14) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), (3.0 - sqrt(5.0)), (fma(0.5, t_0, 1.0) * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(4.0 / Float64(3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -1.5e-5)
		tmp = t_1;
	elseif (x <= 1.38e-14)
		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)), Float64(3.0 - sqrt(5.0)), Float64(fma(0.5, t_0, 1.0) * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -1.5e-5], t$95$1, If[LessEqual[x, 1.38e-14], 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] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000004e-5 or 1.38000000000000002e-14 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-+.f64 58.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{9 \cdot 9 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {\left(\sqrt{5}\right)}^{2} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      8. sqrt-pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{1} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      14. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {\left(\sqrt{5}\right)}^{2}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      15. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{\left(\frac{2}{2}\right)}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{1}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot 5}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 25}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      23. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {\left(\sqrt{5}\right)}^{2}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      24. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{\left(\frac{2}{2}\right)}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{1}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

   8. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{4}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -1.50000000000000004e-5 < x < 1.38000000000000002e-14

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.3%

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

Alternative 20: 78.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, \frac{4}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (fma t_0 (cos x) (/ 4.0 (+ 3.0 (sqrt 5.0)))) 1.0))
          0.3333333333333333)))
   (if (<= x -1.5e-5)
     t_1
     (if (<= x 1.38e-14)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) t_0) 1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), (4.0 / (3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -1.5e-5) {
		tmp = t_1;
	} else if (x <= 1.38e-14) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(4.0 / Float64(3.0 + sqrt(5.0)))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -1.5e-5)
		tmp = t_1;
	elseif (x <= 1.38e-14)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -1.5e-5], t$95$1, If[LessEqual[x, 1.38e-14], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000004e-5 or 1.38000000000000002e-14 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-+.f64 58.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

   6. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{9 \cdot 9 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - \left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {\left(\sqrt{5}\right)}^{2} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      8. sqrt-pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - {5}^{1} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      14. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {\left(\sqrt{5}\right)}^{2}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      15. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{\left(\frac{2}{2}\right)}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot {5}^{1}}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 5 \cdot 5}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      18. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{81 - 25}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      22. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + \sqrt{5} \cdot \sqrt{5}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      23. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {\left(\sqrt{5}\right)}^{2}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      24. sqrt-pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{\left(\frac{2}{2}\right)}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

      25. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{\frac{56}{9 + {5}^{1}}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

   8. Applied rewrites 58.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{4}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

   if -1.50000000000000004e-5 < x < 1.38000000000000002e-14

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.1%

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

Alternative 21: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, 3 + \left(t\_0 \cdot \cos x - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
            (* (- (cos x) 1.0) (sqrt 2.0))
            2.0)
           (fma 0.5 (+ 3.0 (- (* t_0 (cos x)) (sqrt 5.0))) 1.0))
          0.3333333333333333)))
   (if (<= x -1.5e-5)
     t_1
     (if (<= x 1.38e-14)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) t_0) 1.0))
        0.3333333333333333)
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, (3.0 + ((t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -1.5e-5) {
		tmp = t_1;
	} else if (x <= 1.38e-14) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(0.5, Float64(3.0 + Float64(Float64(t_0 * cos(x)) - sqrt(5.0))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -1.5e-5)
		tmp = t_1;
	elseif (x <= 1.38e-14)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(3.0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -1.5e-5], t$95$1, If[LessEqual[x, 1.38e-14], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000004e-5 or 1.38000000000000002e-14 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift-cos.f64 58.4

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

   6. Applied rewrites 58.4%

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

   if -1.50000000000000004e-5 < x < 1.38000000000000002e-14

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.1%

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

Alternative 22: 78.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625, t\_1 \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right), \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), t\_1, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (- (cos x) 1.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -1.5e-5)
     (*
      (/
       (fma -0.0625 (* t_1 (- 0.5 (* (cos (+ x x)) 0.5))) 2.0)
       (fma 0.5 (fma t_0 (cos x) t_2) 1.0))
      0.3333333333333333)
     (if (<= x 1.38e-14)
       (*
        (/
         (fma
          (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 y)))))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0)
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0))
        0.3333333333333333)
       (*
        (/
         (fma (* -0.0625 (- 0.5 (* 0.5 (cos (* 2.0 x))))) t_1 2.0)
         (fma 0.5 (- (fma t_0 (cos x) 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (cos(x) - 1.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -1.5e-5) {
		tmp = (fma(-0.0625, (t_1 * (0.5 - (cos((x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_2), 1.0)) * 0.3333333333333333;
	} else if (x <= 1.38e-14) {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * y))))), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (fma((-0.0625 * (0.5 - (0.5 * cos((2.0 * x))))), t_1, 2.0) / fma(0.5, (fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -1.5e-5)
		tmp = Float64(Float64(fma(-0.0625, Float64(t_1 * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5))), 2.0) / fma(0.5, fma(t_0, cos(x), t_2), 1.0)) * 0.3333333333333333);
	elseif (x <= 1.38e-14)
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(t_2, cos(y), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))), t_1, 2.0) / fma(0.5, Float64(fma(t_0, cos(x), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-5], N[(N[(N[(-0.0625 * N[(t$95$1 * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.38e-14], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000004e-5

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 59.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 59.4%

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

   if -1.50000000000000004e-5 < x < 1.38000000000000002e-14

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.1%

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

   if 1.38000000000000002e-14 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-cos.f64 57.5

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

   6. Applied rewrites 57.5%

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

Alternative 23: 59.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.3%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

6. Applied rewrites 59.3%

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

Alternative 24: 42.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   2.0
   (fma
    0.5
    (fma
     (- (sqrt 5.0) 1.0)
     (cos x)
     (/ (- 9.0 (* (sqrt 5.0) (sqrt 5.0))) (+ 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), ((9.0 - (sqrt(5.0) * sqrt(5.0))) / (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(Float64(9.0 - Float64(sqrt(5.0) * sqrt(5.0))) / 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[(N[(9.0 - N[(N[Sqrt[5.0], $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

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. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot \frac{1}{3} \]

   7. lower-+.f64 59.2

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

6. Applied rewrites 59.2%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right), \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]

7. Taylor expanded in x around 0

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

9. Applied rewrites 42.6%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \frac{9 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right), 1\right)} \cdot 0.3333333333333333 \]
10. Add Preprocessing

Alternative 25: 42.6% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 42.6%

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

Alternative 26: 40.2% accurate, 316.7× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 59.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.2%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(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))))))