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

Percentage Accurate: 99.3% → 99.3%

Time: 12.7s

Alternatives: 35

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

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))))
  (+
   (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
   (* (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y)) 3.0))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((((3.0 - sqrt(5.0)) / 2.0) * cos(y)) * 3.0));
}

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)))) / (((1.0d0 + (((4.0d0 / (sqrt(5.0d0) + 1.0d0)) / 2.0d0) * cos(x))) * 3.0d0) + ((((3.0d0 - sqrt(5.0d0)) / 2.0d0) * cos(y)) * 3.0d0))
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)))) / (((1.0 + (((4.0 / (Math.sqrt(5.0) + 1.0)) / 2.0) * Math.cos(x))) * 3.0) + ((((3.0 - Math.sqrt(5.0)) / 2.0) * Math.cos(y)) * 3.0));
}

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)))) / (((1.0 + (((4.0 / (math.sqrt(5.0) + 1.0)) / 2.0) * math.cos(x))) * 3.0) + ((((3.0 - math.sqrt(5.0)) / 2.0) * math.cos(y)) * 3.0))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)) * 3.0)))
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)))) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((((3.0 - sqrt(5.0)) / 2.0) * cos(y)) * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.3

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-cos.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 2: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{t\_1}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)} \leq 0.54:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right) - 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (<=
        (/
         (+
          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 (* (/ t_1 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))
        0.54)
     (/
      (*
       0.3333333333333333
       (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0))
      (fma 0.5 (- (fma (cos y) t_0 (sqrt 5.0)) 1.0) 1.0))
     (*
      (/ 2.0 (fma 0.5 (fma t_0 (cos y) (* t_1 (cos x))) 1.0))
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (((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 + ((t_1 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))))) <= 0.54) {
		tmp = (0.3333333333333333 * fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0)) / fma(0.5, (fma(cos(y), t_0, sqrt(5.0)) - 1.0), 1.0);
	} else {
		tmp = (2.0 / fma(0.5, fma(t_0, cos(y), (t_1 * cos(x))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (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(t_1 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y))))) <= 0.54)
		tmp = Float64(Float64(0.3333333333333333 * fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0)) / fma(0.5, Float64(fma(cos(y), t_0, sqrt(5.0)) - 1.0), 1.0));
	else
		tmp = Float64(Float64(2.0 / fma(0.5, fma(t_0, cos(y), Float64(t_1 * cos(x))), 1.0)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[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[(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], 0.54], N[(N[(0.3333333333333333 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y))))) < 0.54000000000000004

   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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-cos.f64 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   10. Applied rewrites 74.0%

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

   if 0.54000000000000004 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y)))))

   1. Initial program 98.7%

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

   2. 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 98.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. lower--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-cos.f64 24.9

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

   7. Applied rewrites 24.9%

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

   9. Applied rewrites 24.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 25.1%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (cos x) (cos y)))
   (* (- (sin x) (* 0.0625 (sin y))) (- (sin y) (* 0.0625 (sin x))))
   2.0)
  (+
   (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
   (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0)))))))

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x)))), 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + (1.5 * (cos(y) * (3.0 - sqrt(5.0)))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.3

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-cos.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in y around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lift-cos.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift--.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 99.3

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-cos.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-cos.f64 N/A

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

3. Applied rewrites 99.3%

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

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%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift--.f64 59.6

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

4. Applied rewrites 59.6%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift--.f64 59.6

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

7. Applied rewrites 59.6%

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

8. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift--.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 N/A

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

7. Applied rewrites 99.3%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.3%

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

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in 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%

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

5. Taylor expanded in x around 0

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

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

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

   2. lower--.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-cos.f64 62.0

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

7. Applied rewrites 62.0%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

10. Applied rewrites 99.2%

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

Alternative 10: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{t\_0}{2}\\ t_2 := \cos x - \cos y\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_2}{\mathsf{fma}\left(\cos x, \frac{t\_3}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_3\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_2, \sin x \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(t\_1 \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (/ t_0 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.06)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_2))
      (+ (* (fma (cos x) (/ t_3 2.0) 1.0) 3.0) (* (* (cos y) t_1) 3.0)))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          t_2))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_3) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_3)))
       (/
        (fma (* (sqrt 2.0) t_2) (* (sin x) (- (sin y) (* 0.0625 (sin x)))) 2.0)
        (+
         (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
         (* (* t_1 (cos y)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = t_0 / 2.0;
	double t_2 = cos(x) - cos(y);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.06) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_2)) / ((fma(cos(x), (t_3 / 2.0), 1.0) * 3.0) + ((cos(y) * t_1) * 3.0));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * t_2)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, ((-0.75 * (x * x)) * t_3));
	} else {
		tmp = fma((sqrt(2.0) * t_2), (sin(x) * (sin(y) - (0.0625 * sin(x)))), 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((t_1 * cos(y)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(t_0 / 2.0)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.06)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_2)) / Float64(Float64(fma(cos(x), Float64(t_3 / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_1) * 3.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * t_2)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_3)));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_2), Float64(sin(x) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(t_1 * cos(y)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.06], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.059999999999999998

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lift-sin.f64 64.7

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

   6. Applied rewrites 64.7%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-cos.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in y around 0

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

       1. lift-sin.f64 63.4

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

   11. Applied rewrites 63.4%

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

Alternative 11: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{\mathsf{fma}\left(\cos x, \frac{t\_2}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{t\_0}{2}\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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) (/ (sin x) 16.0))) t_1))
          (+
           (* (fma (cos x) (/ t_2 2.0) 1.0) 3.0)
           (* (* (cos y) (/ t_0 2.0)) 3.0)))))
   (if (<= x -0.06)
     t_3
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          t_1))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       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) - (sin(x) / 16.0))) * t_1)) / ((fma(cos(x), (t_2 / 2.0), 1.0) * 3.0) + ((cos(y) * (t_0 / 2.0)) * 3.0));
	double tmp;
	if (x <= -0.06) {
		tmp = t_3;
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} 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) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(Float64(fma(cos(x), Float64(t_2 / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * Float64(t_0 / 2.0)) * 3.0)))
	tmp = 0.0
	if (x <= -0.06)
		tmp = t_3;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = t_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[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.06], t$95$3, If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.059999999999999998 or 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.9

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

   6. Applied rewrites 63.9%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 12: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := 2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{t\_2}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_3\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{3 \cdot \left(\left(1 + \frac{t\_3}{2} \cdot \cos x\right) + \frac{t\_0}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (+
          2.0
          (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_1)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.06)
     (/ t_2 (* 3.0 (fma 0.5 (fma t_3 (cos x) (* t_0 (cos y))) 1.0)))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          t_1))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_3) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_3)))
       (/
        t_2
        (*
         3.0
         (+ (+ 1.0 (* (/ t_3 2.0) (cos x))) (* (/ t_0 2.0) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = 2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_1);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.06) {
		tmp = t_2 / (3.0 * fma(0.5, fma(t_3, cos(x), (t_0 * cos(y))), 1.0));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, ((-0.75 * (x * x)) * t_3));
	} else {
		tmp = t_2 / (3.0 * ((1.0 + ((t_3 / 2.0) * cos(x))) + ((t_0 / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.06)
		tmp = Float64(t_2 / Float64(3.0 * fma(0.5, fma(t_3, cos(x), Float64(t_0 * cos(y))), 1.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_3), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_3)));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(t_3 / 2.0) * cos(x))) + Float64(Float64(t_0 / 2.0) * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.06], N[(t$95$2 / N[(3.0 * N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.059999999999999998

   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 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 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 64.6

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

   7. Applied rewrites 64.6%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.3

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

   4. Applied rewrites 63.3%

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

Alternative 13: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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) (/ (sin x) 16.0))) t_1))
          (* 3.0 (fma 0.5 (fma t_2 (cos x) (* t_0 (cos y))) 1.0)))))
   (if (<= x -0.06)
     t_3
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          t_1))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_2) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_2)))
       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) - (sin(x) / 16.0))) * t_1)) / (3.0 * fma(0.5, fma(t_2, cos(x), (t_0 * cos(y))), 1.0));
	double tmp;
	if (x <= -0.06) {
		tmp = t_3;
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, ((-0.75 * (x * x)) * t_2));
	} 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) * sin(x)) * Float64(sin(y) - 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)))
	tmp = 0.0
	if (x <= -0.06)
		tmp = t_3;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_0, cos(y), t_2), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_2)));
	else
		tmp = t_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[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - 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]}, If[LessEqual[x, -0.06], t$95$3, If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.059999999999999998 or 1.1600000000000001e-7 < 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 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 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.9

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

   7. Applied rewrites 63.9%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 14: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (*
          (/
           (fma
            (* (sqrt 2.0) t_1)
            (* (- (sin y) (* 0.0625 (sin x))) (sin x))
            2.0)
           (fma 0.5 (fma t_0 (cos x) (* t_2 (cos y))) 1.0))
          0.3333333333333333)))
   (if (<= x -0.06)
     t_3
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          t_1))
        (fma
         (fma 0.5 (fma t_2 (cos y) t_0) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_0)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = (fma((sqrt(2.0) * t_1), ((sin(y) - (0.0625 * sin(x))) * sin(x)), 2.0) / fma(0.5, fma(t_0, cos(x), (t_2 * cos(y))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -0.06) {
		tmp = t_3;
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, ((-0.75 * (x * x)) * t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(fma(Float64(sqrt(2.0) * t_1), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sin(x)), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(t_2 * cos(y))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -0.06)
		tmp = t_3;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * t_1)) / fma(fma(0.5, fma(t_2, cos(y), t_0), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -0.06], t$95$3, If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.059999999999999998 or 1.1600000000000001e-7 < 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 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 98.9%

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

   5. Taylor expanded in y around 0

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

      1. lift-sin.f64 64.0

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

   7. Applied rewrites 64.0%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 15: 81.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;y \leq -0.00072:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{t\_0}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{t\_1}{2}\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* (sqrt 2.0) (- (cos x) (cos y)))
            (* (sin y) (- (sin x) (* 0.0625 (sin y))))
            2.0)
           (fma 0.5 (fma t_0 (cos x) (* t_1 (cos y))) 1.0))
          0.3333333333333333)))
   (if (<= y -0.00072)
     t_2
     (if (<= y 4.8e-13)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+
         (* (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((sqrt(2.0) * (cos(x) - cos(y))), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma(t_0, cos(x), (t_1 * cos(y))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (y <= -0.00072) {
		tmp = t_2;
	} else if (y <= 4.8e-13) {
		tmp = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / ((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 = Float64(Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(t_0, cos(x), Float64(t_1 * cos(y))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (y <= -0.00072)
		tmp = t_2;
	elseif (y <= 4.8e-13)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(Float64(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[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[y, -0.00072], t$95$2, If[LessEqual[y, 4.8e-13], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + 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 y < -7.20000000000000045e-4 or 4.7999999999999997e-13 < 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

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. lift-sin.f64 63.5

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

   7. Applied rewrites 63.5%

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

   if -7.20000000000000045e-4 < y < 4.7999999999999997e-13

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 16: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_4 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot t\_3\right)}{\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_4\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right), 3, \left(-0.75 \cdot \left(x \cdot x\right)\right) \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_3, 2\right)}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(t\_4 \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_4 (/ t_0 2.0)))
   (if (<= x -0.06)
     (/
      (+ 2.0 (* -0.0625 (* t_2 t_3)))
      (+ (* (fma (cos x) (/ t_1 2.0) 1.0) 3.0) (* (* (cos y) t_4) 3.0)))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (* 0.0625 x)))
          (- (cos x) (cos y))))
        (fma
         (fma 0.5 (fma t_0 (cos y) t_1) 1.0)
         3.0
         (* (* -0.75 (* x x)) t_1)))
       (/
        (fma (* -0.0625 t_2) t_3 2.0)
        (+
         (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
         (* (* t_4 (cos y)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = sqrt(2.0) * (cos(x) - 1.0);
	double t_4 = t_0 / 2.0;
	double tmp;
	if (x <= -0.06) {
		tmp = (2.0 + (-0.0625 * (t_2 * t_3))) / ((fma(cos(x), (t_1 / 2.0), 1.0) * 3.0) + ((cos(y) * t_4) * 3.0));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, ((-0.75 * (x * x)) * t_1));
	} else {
		tmp = fma((-0.0625 * t_2), t_3, 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((t_4 * cos(y)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_4 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -0.06)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_3))) / Float64(Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_4) * 3.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(cos(x) - cos(y)))) / fma(fma(0.5, fma(t_0, cos(y), t_1), 1.0), 3.0, Float64(Float64(-0.75 * Float64(x * x)) * t_1)));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_2), t_3, 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(t_4 * cos(y)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.06], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], 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 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.059999999999999998

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-cos.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 61.3

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

   6. Applied rewrites 61.3%

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

   if -0.059999999999999998 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-cos.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.1

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

   8. Applied rewrites 60.1%

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

Alternative 17: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_4 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -0.0004:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot t\_3\right)}{\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_4\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, {\sin y}^{2}, x \cdot \left(1.00390625 \cdot \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_3, 2\right)}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(t\_4 \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_4 (/ t_0 2.0)))
   (if (<= x -0.0004)
     (/
      (+ 2.0 (* -0.0625 (* t_2 t_3)))
      (+ (* (fma (cos x) (/ t_1 2.0) 1.0) 3.0) (* (* (cos y) t_4) 3.0)))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (*
           (sqrt 2.0)
           (fma -0.0625 (pow (sin y) 2.0) (* x (* 1.00390625 (sin y)))))
          (- (cos x) (cos y))))
        (* 3.0 (fma 0.5 (fma t_0 (cos y) t_1) 1.0)))
       (/
        (fma (* -0.0625 t_2) t_3 2.0)
        (+
         (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
         (* (* t_4 (cos y)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = sqrt(2.0) * (cos(x) - 1.0);
	double t_4 = t_0 / 2.0;
	double tmp;
	if (x <= -0.0004) {
		tmp = (2.0 + (-0.0625 * (t_2 * t_3))) / ((fma(cos(x), (t_1 / 2.0), 1.0) * 3.0) + ((cos(y) * t_4) * 3.0));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + ((sqrt(2.0) * fma(-0.0625, pow(sin(y), 2.0), (x * (1.00390625 * sin(y))))) * (cos(x) - cos(y)))) / (3.0 * fma(0.5, fma(t_0, cos(y), t_1), 1.0));
	} else {
		tmp = fma((-0.0625 * t_2), t_3, 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((t_4 * cos(y)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_4 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -0.0004)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_3))) / Float64(Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_4) * 3.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * fma(-0.0625, (sin(y) ^ 2.0), Float64(x * Float64(1.00390625 * sin(y))))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(0.5, fma(t_0, cos(y), t_1), 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_2), t_3, 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(t_4 * cos(y)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.0004], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + N[(x * N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000019e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 61.4

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

   6. Applied rewrites 61.4%

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

   if -4.00000000000000019e-4 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sin.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-cos.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.1

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

   8. Applied rewrites 60.1%

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

Alternative 18: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := {\sin x}^{2}\\ t_3 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_4 := \frac{t\_0}{2}\\ \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_2 \cdot t\_3\right)}{\mathsf{fma}\left(\cos x, \frac{t\_1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_4\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, t\_3, 2\right)}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(t\_4 \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (pow (sin x) 2.0))
        (t_3 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_4 (/ t_0 2.0)))
   (if (<= x -0.00038)
     (/
      (+ 2.0 (* -0.0625 (* t_2 t_3)))
      (+ (* (fma (cos x) (/ t_1 2.0) 1.0) 3.0) (* (* (cos y) t_4) 3.0)))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* (- (sin x) (* 0.0625 (sin y))) (sin y)))
          (- (cos x) (cos y))))
        (* 3.0 (fma 0.5 (fma t_0 (cos y) t_1) 1.0)))
       (/
        (fma (* -0.0625 t_2) t_3 2.0)
        (+
         (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
         (* (* t_4 (cos y)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = sqrt(2.0) * (cos(x) - 1.0);
	double t_4 = t_0 / 2.0;
	double tmp;
	if (x <= -0.00038) {
		tmp = (2.0 + (-0.0625 * (t_2 * t_3))) / ((fma(cos(x), (t_1 / 2.0), 1.0) * 3.0) + ((cos(y) * t_4) * 3.0));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + ((sqrt(2.0) * ((sin(x) - (0.0625 * sin(y))) * sin(y))) * (cos(x) - cos(y)))) / (3.0 * fma(0.5, fma(t_0, cos(y), t_1), 1.0));
	} else {
		tmp = fma((-0.0625 * t_2), t_3, 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((t_4 * cos(y)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = sin(x) ^ 2.0
	t_3 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_4 = Float64(t_0 / 2.0)
	tmp = 0.0
	if (x <= -0.00038)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_2 * t_3))) / Float64(Float64(fma(cos(x), Float64(t_1 / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_4) * 3.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(0.5, fma(t_0, cos(y), t_1), 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_2), t_3, 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(t_4 * cos(y)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.00038], N[(N[(2.0 + N[(-0.0625 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$4), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$4 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000002e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 61.4

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

   6. Applied rewrites 61.4%

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

   if -3.8000000000000002e-4 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. lift-sin.f64 99.3

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

   10. Applied rewrites 99.3%

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

   if 1.1600000000000001e-7 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-cos.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.1

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

   8. Applied rewrites 60.1%

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

Alternative 19: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_2 := \frac{3 - \sqrt{5}}{2}\\ t_3 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_2\right) \cdot 3\\ \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_0 \cdot t\_1\right)}{t\_3}\\ \mathbf{elif}\;x \leq 54:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_0, t\_1, 2\right)}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(t\_2 \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_2 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_3
         (+
          (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0)
          (* (* (cos y) t_2) 3.0))))
   (if (<= x -0.00085)
     (/ (+ 2.0 (* -0.0625 (* t_0 t_1))) t_3)
     (if (<= x 54.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        t_3)
       (/
        (fma (* -0.0625 t_0) t_1 2.0)
        (+
         (* (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x))) 3.0)
         (* (* t_2 (cos y)) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(2.0) * (cos(x) - 1.0);
	double t_2 = (3.0 - sqrt(5.0)) / 2.0;
	double t_3 = (fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + ((cos(y) * t_2) * 3.0);
	double tmp;
	if (x <= -0.00085) {
		tmp = (2.0 + (-0.0625 * (t_0 * t_1))) / t_3;
	} else if (x <= 54.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_3;
	} else {
		tmp = fma((-0.0625 * t_0), t_1, 2.0) / (((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + ((t_2 * cos(y)) * 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_2 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_3 = Float64(Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_2) * 3.0))
	tmp = 0.0
	if (x <= -0.00085)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * t_1))) / t_3);
	elseif (x <= 54.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_3);
	else
		tmp = Float64(fma(Float64(-0.0625 * t_0), t_1, 2.0) / Float64(Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) * 3.0) + Float64(Float64(t_2 * cos(y)) * 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00085], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[x, 54.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$0), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999953e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 61.3

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

   6. Applied rewrites 61.3%

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

   if -8.49999999999999953e-4 < x < 54

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 54 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-cos.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-cos.f64 60.2

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

   8. Applied rewrites 60.2%

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

Alternative 20: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_1 := \frac{3 - \sqrt{5}}{2}\\ t_2 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot t\_1\right) \cdot 3\\ \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;\frac{t\_0}{t\_2}\\ \mathbf{elif}\;x \leq 54:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
        (t_1 (/ (- 3.0 (sqrt 5.0)) 2.0))
        (t_2
         (+
          (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0)
          (* (* (cos y) t_1) 3.0))))
   (if (<= x -0.00085)
     (/ t_0 t_2)
     (if (<= x 54.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        t_2)
       (/
        t_0
        (*
         3.0
         (+
          (+ 1.0 (* (/ (/ 4.0 (+ (sqrt 5.0) 1.0)) 2.0) (cos x)))
          (* t_1 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_1 = (3.0 - sqrt(5.0)) / 2.0;
	double t_2 = (fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + ((cos(y) * t_1) * 3.0);
	double tmp;
	if (x <= -0.00085) {
		tmp = t_0 / t_2;
	} else if (x <= 54.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_2;
	} else {
		tmp = t_0 / (3.0 * ((1.0 + (((4.0 / (sqrt(5.0) + 1.0)) / 2.0) * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_1 = Float64(Float64(3.0 - sqrt(5.0)) / 2.0)
	t_2 = Float64(Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * t_1) * 3.0))
	tmp = 0.0
	if (x <= -0.00085)
		tmp = Float64(t_0 / t_2);
	elseif (x <= 54.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_2);
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(4.0 / Float64(sqrt(5.0) + 1.0)) / 2.0) * cos(x))) + Float64(t_1 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00085], N[(t$95$0 / t$95$2), $MachinePrecision], If[LessEqual[x, 54.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(N[(1.0 + N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * 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 x < -8.49999999999999953e-4

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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 98.9%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift--.f64 61.3

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

   6. Applied rewrites 61.3%

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

   if -8.49999999999999953e-4 < x < 54

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-cos.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 54 < x

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f64 98.9

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

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift--.f64 60.2

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Applied rewrites 60.2%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 54:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0)
          (* (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) 3.0)))
        (t_1
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          t_0)))
   (if (<= x -0.00085)
     t_1
     (if (<= x 54.0)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = (fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) * 3.0);
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_0;
	double tmp;
	if (x <= -0.00085) {
		tmp = t_1;
	} else if (x <= 54.0) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) * 3.0))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / t_0)
	tmp = 0.0
	if (x <= -0.00085)
		tmp = t_1;
	elseif (x <= 54.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.00085], t$95$1, If[LessEqual[x, 54.0], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999953e-4 or 54 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      8. lift--.f64 60.7

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

   6. Applied rewrites 60.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

   if -8.49999999999999953e-4 < x < 54

   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%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      8. lift-cos.f64 98.5

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

   6. Applied rewrites 98.5%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{t\_0}{2}\right) \cdot 3}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (+
           2.0
           (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
          (+
           (* (fma (cos x) (/ (- (sqrt 5.0) 1.0) 2.0) 1.0) 3.0)
           (* (* (cos y) (/ t_0 2.0)) 3.0)))))
   (if (<= x -8e-6)
     t_1
     (if (<= x 1.16e-7)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* 3.0 (+ (/ 2.0 (+ (sqrt 5.0) 1.0)) 1.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / ((fma(cos(x), ((sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + ((cos(y) * (t_0 / 2.0)) * 3.0));
	double tmp;
	if (x <= -8e-6) {
		tmp = t_1;
	} else if (x <= 1.16e-7) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_0, (3.0 * ((2.0 / (sqrt(5.0) + 1.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(Float64(fma(cos(x), Float64(Float64(sqrt(5.0) - 1.0) / 2.0), 1.0) * 3.0) + Float64(Float64(cos(y) * Float64(t_0 / 2.0)) * 3.0)))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = t_1;
	elseif (x <= 1.16e-7)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(3.0 * Float64(Float64(2.0 / Float64(sqrt(5.0) + 1.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] / 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], t$95$1, If[LessEqual[x, 1.16e-7], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-6 or 1.1600000000000001e-7 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5}} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

      8. lift--.f64 60.7

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

   6. Applied rewrites 60.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3} \]

   if -7.99999999999999964e-6 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5}} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \color{blue}{\sqrt{5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5 \cdot 5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{5 - \color{blue}{1}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{2 \cdot 2}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{2 \cdot 2}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5}} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_3, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_2 \cdot \cos y\right), 1\right)}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_1 \cdot t\_3\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_0 \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* (sqrt 2.0) (- (cos x) 1.0))))
   (if (<= x -8e-6)
     (/
      (fma (* -0.0625 t_1) t_3 2.0)
      (* 3.0 (fma 0.5 (fma t_0 (cos x) (* t_2 (cos y))) 1.0)))
     (if (<= x 1.16e-7)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_2 (* 3.0 (+ (/ 2.0 (+ (sqrt 5.0) 1.0)) 1.0))))
       (*
        (/
         (- 2.0 (* 0.0625 (* t_1 t_3)))
         (fma 0.5 (fma t_2 (cos y) (* t_0 (cos x))) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = sqrt(2.0) * (cos(x) - 1.0);
	double tmp;
	if (x <= -8e-6) {
		tmp = fma((-0.0625 * t_1), t_3, 2.0) / (3.0 * fma(0.5, fma(t_0, cos(x), (t_2 * cos(y))), 1.0));
	} else if (x <= 1.16e-7) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_2, (3.0 * ((2.0 / (sqrt(5.0) + 1.0)) + 1.0)));
	} else {
		tmp = ((2.0 - (0.0625 * (t_1 * t_3))) / fma(0.5, fma(t_2, cos(y), (t_0 * cos(x))), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(fma(Float64(-0.0625 * t_1), t_3, 2.0) / Float64(3.0 * fma(0.5, fma(t_0, cos(x), Float64(t_2 * cos(y))), 1.0)));
	elseif (x <= 1.16e-7)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_2, Float64(3.0 * Float64(Float64(2.0 / Float64(sqrt(5.0) + 1.0)) + 1.0))));
	else
		tmp = Float64(Float64(Float64(2.0 - Float64(0.0625 * Float64(t_1 * t_3))) / fma(0.5, fma(t_2, cos(y), Float64(t_0 * cos(x))), 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(3.0 * N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[(0.0625 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in 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 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2} \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      10. lift-cos.f64 61.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

   7. Applied rewrites 61.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

   if -7.99999999999999964e-6 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5}} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \color{blue}{\sqrt{5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5 \cdot 5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{5 - \color{blue}{1}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{2 \cdot 2}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{2 \cdot 2}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5}} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}} \]

   if 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   9. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]
   11. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       3. metadata-eval N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       6. lower-pow.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       7. lift-sin.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       9. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]

       11. lift-cos.f64 60.1

           \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333 \]

   12. Applied rewrites 60.1%

       \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right), 1\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* -0.0625 (pow (sin x) 2.0))
           (* (sqrt 2.0) (- (cos x) 1.0))
           2.0)
          (*
           3.0
           (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) 1.0)))))
   (if (<= x -8e-6)
     t_1
     (if (<= x 1.16e-7)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* 3.0 (+ (/ 2.0 (+ (sqrt 5.0) 1.0)) 1.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((-0.0625 * pow(sin(x), 2.0)), (sqrt(2.0) * (cos(x) - 1.0)), 2.0) / (3.0 * fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))), 1.0));
	double tmp;
	if (x <= -8e-6) {
		tmp = t_1;
	} else if (x <= 1.16e-7) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_0, (3.0 * ((2.0 / (sqrt(5.0) + 1.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(sqrt(2.0) * Float64(cos(x) - 1.0)), 2.0) / Float64(3.0 * fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))), 1.0)))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = t_1;
	elseif (x <= 1.16e-7)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(3.0 * Float64(Float64(2.0 / Float64(sqrt(5.0) + 1.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], t$95$1, If[LessEqual[x, 1.16e-7], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-6 or 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + \color{blue}{2}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2} \cdot \left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

      10. lift-cos.f64 60.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

   7. Applied rewrites 60.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \]

   if -7.99999999999999964e-6 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5}} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \color{blue}{\sqrt{5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5 \cdot 5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{5 - \color{blue}{1}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{2 \cdot 2}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{2 \cdot 2}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5}} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (pow (sin x) 2.0))
            (* (sqrt 2.0) (- (cos x) 1.0))
            2.0)
           (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) 1.0))
          0.3333333333333333)))
   (if (<= x -8e-6)
     t_1
     (if (<= x 1.16e-7)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* 3.0 (+ (/ 2.0 (+ (sqrt 5.0) 1.0)) 1.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((-0.0625 * pow(sin(x), 2.0)), (sqrt(2.0) * (cos(x) - 1.0)), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -8e-6) {
		tmp = t_1;
	} else if (x <= 1.16e-7) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_0, (3.0 * ((2.0 / (sqrt(5.0) + 1.0)) + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(sqrt(2.0) * Float64(cos(x) - 1.0)), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -8e-6)
		tmp = t_1;
	elseif (x <= 1.16e-7)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(3.0 * Float64(Float64(2.0 / Float64(sqrt(5.0) + 1.0)) + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -8e-6], t$95$1, If[LessEqual[x, 1.16e-7], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-6 or 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.1

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.1%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) + 2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lift-cos.f64 60.7

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   10. Applied rewrites 60.7%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   if -7.99999999999999964e-6 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5}} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \color{blue}{\sqrt{5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5 \cdot 5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{5 - \color{blue}{1}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{2 \cdot 2}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{2 \cdot 2}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5}} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_1 \cdot t\_2\right)}{\mathsf{fma}\left(1.5, t\_0, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_2, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_3, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -6.8e-5)
     (/
      (- 2.0 (* 0.0625 (* t_1 t_2)))
      (fma 1.5 t_0 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_3))))))
     (if (<= x 1.16e-7)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (fma (* 1.5 (cos y)) t_0 (* 3.0 (+ (/ 2.0 (+ (sqrt 5.0) 1.0)) 1.0))))
       (*
        (/
         (fma (* -0.0625 t_1) t_2 2.0)
         (fma 0.5 (- (fma (cos x) t_3 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(2.0) * (cos(x) - 1.0);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = (2.0 - (0.0625 * (t_1 * t_2))) / fma(1.5, t_0, (3.0 * (1.0 - (-0.5 * (cos(x) * t_3)))));
	} else if (x <= 1.16e-7) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / fma((1.5 * cos(y)), t_0, (3.0 * ((2.0 / (sqrt(5.0) + 1.0)) + 1.0)));
	} else {
		tmp = (fma((-0.0625 * t_1), t_2, 2.0) / fma(0.5, (fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_1 * t_2))) / fma(1.5, t_0, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_3))))));
	elseif (x <= 1.16e-7)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / fma(Float64(1.5 * cos(y)), t_0, Float64(3.0 * Float64(Float64(2.0 / Float64(sqrt(5.0) + 1.0)) + 1.0))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), t_2, 2.0) / fma(0.5, Float64(fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], N[(N[(2.0 - N[(0.0625 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$0 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(3.0 * N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 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 < -6.7999999999999999e-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. 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 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{\sqrt{5} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5}} \cdot \sqrt{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{5} \cdot \color{blue}{\sqrt{5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{\sqrt{5 \cdot 5}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\sqrt{\color{blue}{25}} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{5} - 1 \cdot 1}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{5 - \color{blue}{1}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{2 \cdot 2}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{2 \cdot 2}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{\color{blue}{4}}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\frac{4}{\sqrt{5} + 1}}{2}} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\color{blue}{\frac{4}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5} + 1}}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\color{blue}{\sqrt{5}} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \color{blue}{\cos x}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{\color{blue}{3 - \sqrt{5}}}{2} \cdot \cos y\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \color{blue}{\sqrt{5}}}{2} \cdot \cos y\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \color{blue}{\cos y}\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\frac{4}{\sqrt{5} + 1}}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + 2 \cdot \frac{1}{1 + \sqrt{5}}\right)}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{2}{\sqrt{5} + 1} + 1\right)\right)}} \]

   if 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 58.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_1 \cdot t\_2\right)}{\mathsf{fma}\left(1.5, t\_0, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot t\_0, 3 \cdot \left(1 + 0.5 \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_2, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_3, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -6.8e-5)
     (/
      (- 2.0 (* 0.0625 (* t_1 t_2)))
      (fma 1.5 t_0 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_3))))))
     (if (<= x 1.16e-7)
       (/
        (- 2.0 (* 0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (fma 1.5 (* (cos y) t_0) (* 3.0 (+ 1.0 (* 0.5 t_3)))))
       (*
        (/
         (fma (* -0.0625 t_1) t_2 2.0)
         (fma 0.5 (- (fma (cos x) t_3 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(2.0) * (cos(x) - 1.0);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = (2.0 - (0.0625 * (t_1 * t_2))) / fma(1.5, t_0, (3.0 * (1.0 - (-0.5 * (cos(x) * t_3)))));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 - (0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / fma(1.5, (cos(y) * t_0), (3.0 * (1.0 + (0.5 * t_3))));
	} else {
		tmp = (fma((-0.0625 * t_1), t_2, 2.0) / fma(0.5, (fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_1 * t_2))) / fma(1.5, t_0, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_3))))));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / fma(1.5, Float64(cos(y) * t_0), Float64(3.0 * Float64(1.0 + Float64(0.5 * t_3)))));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), t_2, 2.0) / fma(0.5, Float64(fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], N[(N[(2.0 - N[(0.0625 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$0 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(2.0 - N[(0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 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 < -6.7999999999999999e-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. 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 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. 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%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(1.5, \cos y \cdot \left(3 - \sqrt{5}\right), 3 \cdot \left(1 + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   if 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 58.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot \left(\cos x - 1\right)\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - 0.0625 \cdot \left(t\_1 \cdot t\_2\right)}{\mathsf{fma}\left(1.5, t\_0, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot t\_3\right)\right)\right)}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_3\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, t\_2, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_3, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 2.0) (- (cos x) 1.0)))
        (t_3 (- (sqrt 5.0) 1.0)))
   (if (<= x -6.8e-5)
     (/
      (- 2.0 (* 0.0625 (* t_1 t_2)))
      (fma 1.5 t_0 (* 3.0 (- 1.0 (* -0.5 (* (cos x) t_3))))))
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* 3.0 (fma 0.5 (fma t_0 (cos y) t_3) 1.0)))
       (*
        (/
         (fma (* -0.0625 t_1) t_2 2.0)
         (fma 0.5 (- (fma (cos x) t_3 3.0) (sqrt 5.0)) 1.0))
        0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = sqrt(2.0) * (cos(x) - 1.0);
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = (2.0 - (0.0625 * (t_1 * t_2))) / fma(1.5, t_0, (3.0 * (1.0 - (-0.5 * (cos(x) * t_3)))));
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * fma(0.5, fma(t_0, cos(y), t_3), 1.0));
	} else {
		tmp = (fma((-0.0625 * t_1), t_2, 2.0) / fma(0.5, (fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(2.0) * Float64(cos(x) - 1.0))
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = Float64(Float64(2.0 - Float64(0.0625 * Float64(t_1 * t_2))) / fma(1.5, t_0, Float64(3.0 * Float64(1.0 - Float64(-0.5 * Float64(cos(x) * t_3))))));
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * fma(0.5, fma(t_0, cos(y), t_3), 1.0)));
	else
		tmp = Float64(Float64(fma(Float64(-0.0625 * t_1), t_2, 2.0) / fma(0.5, Float64(fma(cos(x), t_3, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], N[(N[(2.0 - N[(0.0625 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * t$95$0 + N[(3.0 * N[(1.0 - N[(-0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.16e-7], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 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 < -6.7999999999999999e-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. 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 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \frac{\sqrt{5} - 1}{2}, 1\right) \cdot 3 + \left(\cos y \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot 3}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{2 - 0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3 \cdot \left(1 - -0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)}, 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \left(3 - \sqrt{5}\right) \cdot \cos y + \left(\color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos \color{blue}{y}, \sqrt{5} - 1\right), 1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      10. lift--.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      8. lift-cos.f64 99.2

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   if 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 58.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (pow (sin x) 2.0))
            (* (sqrt 2.0) (- (cos x) 1.0))
            2.0)
           (fma 0.5 (- (fma (cos x) t_0 3.0) (sqrt 5.0)) 1.0))
          0.3333333333333333)))
   (if (<= x -6.8e-5)
     t_1
     (if (<= x 1.16e-7)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (* 3.0 (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) t_0) 1.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (fma((-0.0625 * pow(sin(x), 2.0)), (sqrt(2.0) * (cos(x) - 1.0)), 2.0) / fma(0.5, (fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = t_1;
	} else if (x <= 1.16e-7) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), t_0), 1.0));
	} 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 * (sin(x) ^ 2.0)), Float64(sqrt(2.0) * Float64(cos(x) - 1.0)), 2.0) / fma(0.5, Float64(fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = t_1;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), t_0), 1.0)));
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], t$95$1, If[LessEqual[x, 1.16e-7], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999999e-5 or 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{1}\right)} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)}, 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \left(3 - \sqrt{5}\right) \cdot \cos y + \left(\color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos \color{blue}{y}, \sqrt{5} - 1\right), 1\right)} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      10. lift--.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      8. lift-cos.f64 99.2

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (*
          (/
           (fma
            (* -0.0625 (pow (sin x) 2.0))
            (* (sqrt 2.0) (- (cos x) 1.0))
            2.0)
           (fma 0.5 (- (fma (cos x) t_0 3.0) (sqrt 5.0)) 1.0))
          0.3333333333333333)))
   (if (<= x -6.8e-5)
     t_1
     (if (<= x 1.16e-7)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 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 * pow(sin(x), 2.0)), (sqrt(2.0) * (cos(x) - 1.0)), 2.0) / fma(0.5, (fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = t_1;
	} else if (x <= 1.16e-7) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((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 * (sin(x) ^ 2.0)), Float64(sqrt(2.0) * Float64(cos(x) - 1.0)), 2.0) / fma(0.5, Float64(fma(cos(x), t_0, 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = t_1;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], t$95$1, If[LessEqual[x, 1.16e-7], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(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 < -6.7999999999999999e-5 or 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) - 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (fma
            (* -0.0625 (pow (sin x) 2.0))
            (* (sqrt 2.0) (- (cos x) 1.0))
            2.0)
           (fma 0.5 (- (fma (cos x) (- (sqrt 5.0) 1.0) 3.0) (sqrt 5.0)) 1.0))
          0.3333333333333333)))
   (if (<= x -6.8e-5)
     t_0
     (if (<= x 1.16e-7)
       (/
        (*
         0.3333333333333333
         (fma
          (* -0.0625 (pow (sin y) 2.0))
          (* (sqrt 2.0) (- 1.0 (cos y)))
          2.0))
        (fma 0.5 (- (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) 1.0) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (fma((-0.0625 * pow(sin(x), 2.0)), (sqrt(2.0) * (cos(x) - 1.0)), 2.0) / fma(0.5, (fma(cos(x), (sqrt(5.0) - 1.0), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -6.8e-5) {
		tmp = t_0;
	} else if (x <= 1.16e-7) {
		tmp = (0.3333333333333333 * fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0)) / fma(0.5, (fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)) - 1.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(sqrt(2.0) * Float64(cos(x) - 1.0)), 2.0) / fma(0.5, Float64(fma(cos(x), Float64(sqrt(5.0) - 1.0), 3.0) - sqrt(5.0)), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -6.8e-5)
		tmp = t_0;
	elseif (x <= 1.16e-7)
		tmp = Float64(Float64(0.3333333333333333 * fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0)) / fma(0.5, Float64(fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)) - 1.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -6.8e-5], t$95$0, If[LessEqual[x, 1.16e-7], N[(N[(0.3333333333333333 * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 * N[(N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999999e-5 or 1.1600000000000001e-7 < 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 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 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 27.0

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 27.0%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(3 + \cos x \cdot \left(\sqrt{5} - 1\right)\right) - \sqrt{5}\right)} \cdot \frac{1}{3} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3\right) - \sqrt{5}, 1\right)} \cdot 0.3333333333333333 \]

   if -6.7999999999999999e-5 < x < 1.1600000000000001e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 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 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

      11. lift-cos.f64 99.1

          \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   7. Applied rewrites 99.1%

      \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \color{blue}{\frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \color{blue}{\frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right)}} \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) - 1, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 45.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   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 (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(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[(2.0 / 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] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
6. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   3. metadata-eval N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   11. lift-cos.f64 62.0

       \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

7. Applied rewrites 62.0%

   \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
9. Step-by-step derivation

10. Applied rewrites 45.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]
11. Add Preprocessing

Alternative 33: 45.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   2.0
   (fma
    0.5
    (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x)))
    1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (2.0 / fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]
6. Step-by-step derivation

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   3. metadata-eval N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{2 - \frac{1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot \frac{1}{3} \]

   11. lift-cos.f64 62.0

       \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

7. Applied rewrites 62.0%

   \[\leadsto \frac{2 - 0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 1\right)} \cdot 0.3333333333333333 \]

9. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333} \]

10. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]
11. Step-by-step derivation

12. Applied rewrites 45.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333 \]
13. Add Preprocessing

Alternative 34: 43.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/ 2.0 (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (2.0 / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 43.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 35: 40.4% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \color{blue}{\frac{1}{3}} \]

4. Applied rewrites 60.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \]
6. Step-by-step derivation

7. Applied rewrites 40.4%

   \[\leadsto 0.3333333333333333 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025095 
(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))))))