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

Percentage Accurate: 99.3% → 99.3%

Time: 11.4s

Alternatives: 36

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (+
  2
  (*
   (*
    (* (sqrt 2) (- (sin x) (/ (sin y) 16)))
    (- (sin y) (/ (sin x) 16)))
   (- (cos x) (cos y))))
 (*
  3
  (+
   (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x)))
   (* (/ (- 3 (sqrt 5)) 2) (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 + N[(N[(N[(N[Sqrt[2], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3 * N[(N[(1 + N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] / 2), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (+
  2
  (*
   (*
    (* (sqrt 2) (- (sin x) (/ (sin y) 16)))
    (- (sin y) (/ (sin x) 16)))
   (- (cos x) (cos y))))
 (*
  3
  (+
   (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x)))
   (* (/ (- 3 (sqrt 5)) 2) (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 + N[(N[(N[(N[Sqrt[2], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3 * N[(N[(1 + N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] / 2), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (/
  (-
   (*
    (- (cos y) (cos x))
    (*
     (* (- (sin y) (* 1/16 (sin x))) (sqrt 2))
     (- (sin x) (* 1/16 (sin y)))))
   2)
  (-
   (* (* (- (* (sqrt 5) 1/3) 1) 3/2) (cos y))
   (- (* (* 1/2 (- (sqrt 5) 1)) (cos x)) -1)))
 3))

C

double code(double x, double y) {
	return ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - (((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0))) / 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 = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625d0 * sin(x))) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / (((((sqrt(5.0d0) * 0.3333333333333333d0) - 1.0d0) * 1.5d0) * cos(y)) - (((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (-1.0d0)))) / 3.0d0
end function

Java

public static double code(double x, double y) {
	return ((((Math.cos(y) - Math.cos(x)) * (((Math.sin(y) - (0.0625 * Math.sin(x))) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0) / (((((Math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * Math.cos(y)) - (((0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x)) - -1.0))) / 3.0;
}

Python

def code(x, y):
	return ((((math.cos(y) - math.cos(x)) * (((math.sin(y) - (0.0625 * math.sin(x))) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / (((((math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * math.cos(y)) - (((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - -1.0))) / 3.0

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - Float64(Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x)) - -1.0))) / 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - (((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0))) / 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] * 1/3), $MachinePrecision] - 1), $MachinePrecision] * 3/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-/.f64 99.3%

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. mult-flip-rev N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. sub-to-mult-rev N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. distribute-neg-frac N/A

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

   11. lift--.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. div-sub N/A

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

   15. metadata-eval N/A

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

   16. mult-flip-rev N/A

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

   17. metadata-eval N/A

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

   18. lift-*.f64 N/A

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

   19. sub-to-mult-rev N/A

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

7. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (/
  (-
   (*
    (- (cos y) (cos x))
    (*
     (* (- (sin y) (* 1/16 (sin x))) (sqrt 2))
     (- (sin x) (* 1/16 (sin y)))))
   2)
  (-
   (- (* (* (cos y) 1/2) (- (sqrt 5) 3)) 1)
   (* (* (cos x) 1/2) (- (sqrt 5) 1))))
 3))

C

double code(double x, double y) {
	return ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / ((((cos(y) * 0.5) * (sqrt(5.0) - 3.0)) - 1.0) - ((cos(x) * 0.5) * (sqrt(5.0) - 1.0)))) / 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 = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625d0 * sin(x))) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / ((((cos(y) * 0.5d0) * (sqrt(5.0d0) - 3.0d0)) - 1.0d0) - ((cos(x) * 0.5d0) * (sqrt(5.0d0) - 1.0d0)))) / 3.0d0
end function

Java

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

Python

def code(x, y):
	return ((((math.cos(y) - math.cos(x)) * (((math.sin(y) - (0.0625 * math.sin(x))) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / ((((math.cos(y) * 0.5) * (math.sqrt(5.0) - 3.0)) - 1.0) - ((math.cos(x) * 0.5) * (math.sqrt(5.0) - 1.0)))) / 3.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / ((((cos(y) * 0.5) * (sqrt(5.0) - 3.0)) - 1.0) - ((cos(x) * 0.5) * (sqrt(5.0) - 1.0)))) / 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 1/2), $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision] - N[(N[(N[Cos[x], $MachinePrecision] * 1/2), $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (/
  (-
   (*
    (- (cos y) (cos x))
    (*
     (* (- (sin y) (* 1/16 (sin x))) (sqrt 2))
     (- (sin x) (* 1/16 (sin y)))))
   2)
  (-
   (* (* (- (sqrt 5) 3) 1/2) (cos y))
   (- (* (* 1/2 (- (sqrt 5) 1)) (cos x)) -1)))
 3))

C

double code(double x, double y) {
	return ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - (((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0))) / 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 = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625d0 * sin(x))) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / ((((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)) - (((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (-1.0d0)))) / 3.0d0
end function

Java

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

Python

def code(x, y):
	return ((((math.cos(y) - math.cos(x)) * (((math.sin(y) - (0.0625 * math.sin(x))) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / ((((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)) - (((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - -1.0))) / 3.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((((cos(y) - cos(x)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - (((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0))) / 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (*
  (-
   (*
    (- (cos x) (cos y))
    (*
     (* (- (sin y) (* 1/16 (sin x))) (sqrt 2))
     (- (sin x) (* 1/16 (sin y)))))
   -2)
  1/3)
 (-
  (- (* (* 1/2 (- (sqrt 5) 1)) (cos x)) -1)
  (* (* (- (sqrt 5) 3) 1/2) (cos y)))))

C

double code(double x, double y) {
	return ((((cos(x) - cos(y)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - -2.0) * 0.3333333333333333) / ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0) - (((sqrt(5.0) - 3.0) * 0.5) * 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 = ((((cos(x) - cos(y)) * (((sin(y) - (0.0625d0 * sin(x))) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - (-2.0d0)) * 0.3333333333333333d0) / ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (-1.0d0)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)))
end function

Java

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

Python

def code(x, y):
	return ((((math.cos(x) - math.cos(y)) * (((math.sin(y) - (0.0625 * math.sin(x))) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - -2.0) * 0.3333333333333333) / ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - -1.0) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((((cos(x) - cos(y)) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - -2.0) * 0.3333333333333333) / ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - -1.0) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y)));
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(N[(N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.2%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (-
  (*
   (- (cos x) (cos y))
   (*
    (* (- (sin x) (* 1/16 (sin y))) (sqrt 2))
    (- (sin y) (* (sin x) 1/16))))
  -2)
 (/
  1/3
  (-
   (- (* (* (cos x) 1/2) (- (sqrt 5) 1)) -1)
   (* (* (cos y) 1/2) (- (sqrt 5) 3))))))

C

double code(double x, double y) {
	return (((cos(x) - cos(y)) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (sin(y) - (sin(x) * 0.0625)))) - -2.0) * (0.3333333333333333 / ((((cos(x) * 0.5) * (sqrt(5.0) - 1.0)) - -1.0) - ((cos(y) * 0.5) * (sqrt(5.0) - 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 = (((cos(x) - cos(y)) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * (sin(y) - (sin(x) * 0.0625d0)))) - (-2.0d0)) * (0.3333333333333333d0 / ((((cos(x) * 0.5d0) * (sqrt(5.0d0) - 1.0d0)) - (-1.0d0)) - ((cos(y) * 0.5d0) * (sqrt(5.0d0) - 3.0d0))))
end function

Java

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

Python

def code(x, y):
	return (((math.cos(x) - math.cos(y)) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * (math.sin(y) - (math.sin(x) * 0.0625)))) - -2.0) * (0.3333333333333333 / ((((math.cos(x) * 0.5) * (math.sqrt(5.0) - 1.0)) - -1.0) - ((math.cos(y) * 0.5) * (math.sqrt(5.0) - 3.0))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (((cos(x) - cos(y)) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (sin(y) - (sin(x) * 0.0625)))) - -2.0) * (0.3333333333333333 / ((((cos(x) * 0.5) * (sqrt(5.0) - 1.0)) - -1.0) - ((cos(y) * 0.5) * (sqrt(5.0) - 3.0))));
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * N[(1/3 / N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * 1/2), $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(N[Cos[y], $MachinePrecision] * 1/2), $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

5. Applied rewrites 99.2%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (*
 1/3
 (/
  (+
   2
   (*
    (sqrt 2)
    (*
     (- (cos x) (cos y))
     (* (- (sin x) (* 1/16 (sin y))) (- (sin y) (* 1/16 (sin x)))))))
  (+
   1
   (+
    (* 1/2 (* (cos x) (- (sqrt 5) 1)))
    (* 1/2 (* (cos y) (- 3 (sqrt 5)))))))))

C

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

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 = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((sin(x) - (0.0625d0 * sin(y))) * (sin(y) - (0.0625d0 * sin(x))))))) / (1.0d0 + ((0.5d0 * (cos(x) * (sqrt(5.0d0) - 1.0d0))) + (0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
end function

Java

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

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (0.0625 * math.sin(y))) * (math.sin(y) - (0.0625 * math.sin(x))))))) / (1.0 + ((0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))) + (0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (0.0625 * sin(y))) * (sin(y) - (0.0625 * sin(x))))))) / (1.0 + ((0.5 * (cos(x) * (sqrt(5.0) - 1.0))) + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
end

Wolfram

code[x_, y_] := N[(1/3 * N[(N[(2 + N[(N[Sqrt[2], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1 + N[(N[(1/2 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 * N[(N[Cos[y], $MachinePrecision] * N[(3 - N[Sqrt[5], $MachinePrecision]), $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. 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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 59.6%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.2%

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

Alternative 7: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\right)}}{3}\\ \mathbf{if}\;y \leq \frac{-8358680908399641}{288230376151711744}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq \frac{1080863910568919}{72057594037927936}:\\ \;\;\;\;\left(\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}\right) \cdot \frac{-1}{\left(\cos y \cdot \frac{1}{2}\right) \cdot \left(\sqrt{5} - 3\right) - \left(\left(\cos x \cdot \frac{1}{2}\right) \cdot t\_0 - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1
        (/
         (/
          (-
           (*
            (- (cos y) (cos x))
            (* (* (sin y) (sqrt 2)) (- (sin x) (* 1/16 (sin y)))))
           2)
          (-
           (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y))
           (- (* (* 1/2 t_0) (cos x)) -1)))
         3)))
  (if (<= y -8358680908399641/288230376151711744)
    t_1
    (if (<= y 1080863910568919/72057594037927936)
      (*
       (*
        (-
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 y)) (sqrt 2))))
         -2)
        1/3)
       (/
        -1
        (-
         (* (* (cos y) 1/2) (- (sqrt 5) 3))
         (- (* (* (cos x) 1/2) t_0) -1))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (y <= -0.029) {
		tmp = t_1;
	} else if (y <= 0.015) {
		tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((cos(y) * 0.5) * (sqrt(5.0) - 3.0)) - (((cos(x) * 0.5) * t_0) - -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - (((0.5d0 * t_0) * cos(x)) - (-1.0d0)))) / 3.0d0
    if (y <= (-0.029d0)) then
        tmp = t_1
    else if (y <= 0.015d0) then
        tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) * ((-1.0d0) / (((cos(y) * 0.5d0) * (sqrt(5.0d0) - 3.0d0)) - (((cos(x) * 0.5d0) * t_0) - (-1.0d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((((Math.cos(y) - Math.cos(x)) * ((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - (((0.5 * t_0) * Math.cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (y <= -0.029) {
		tmp = t_1;
	} else if (y <= 0.015) {
		tmp = ((((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((Math.cos(y) * 0.5) * (Math.sqrt(5.0) - 3.0)) - (((Math.cos(x) * 0.5) * t_0) - -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((((math.cos(y) - math.cos(x)) * ((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - (((0.5 * t_0) * math.cos(x)) - -1.0))) / 3.0
	tmp = 0
	if y <= -0.029:
		tmp = t_1
	elif y <= 0.015:
		tmp = ((((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((math.cos(y) * 0.5) * (math.sqrt(5.0) - 3.0)) - (((math.cos(x) * 0.5) * 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(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0))) / 3.0)
	tmp = 0.0
	if (y <= -0.029)
		tmp = t_1;
	elseif (y <= 0.015)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * Float64(-1.0 / Float64(Float64(Float64(cos(y) * 0.5) * Float64(sqrt(5.0) - 3.0)) - Float64(Float64(Float64(cos(x) * 0.5) * t_0) - -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	tmp = 0.0;
	if (y <= -0.029)
		tmp = t_1;
	elseif (y <= 0.015)
		tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((cos(y) * 0.5) * (sqrt(5.0) - 3.0)) - (((cos(x) * 0.5) * t_0) - -1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]}, If[LessEqual[y, -8358680908399641/288230376151711744], t$95$1, If[LessEqual[y, 1080863910568919/72057594037927936], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] * N[(-1 / N[(N[(N[(N[Cos[y], $MachinePrecision] * 1/2), $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Cos[x], $MachinePrecision] * 1/2), $MachinePrecision] * t$95$0), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.029000000000000001 or 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   8. Applied rewrites 64.1%

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

   if -0.029000000000000001 < y < 0.014999999999999999

   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

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

      1. lower-*.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.5%

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

Alternative 8: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos y - \cos x\right) \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2\\ t_2 := \sqrt{5} - 3\\ t_3 := \left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\\ \mathbf{if}\;y \leq \frac{-8358680908399641}{288230376151711744}:\\ \;\;\;\;\frac{\frac{t\_1}{\left(t\_2 \cdot \frac{1}{2}\right) \cdot \cos y - t\_3}}{3}\\ \mathbf{elif}\;y \leq \frac{1080863910568919}{72057594037927936}:\\ \;\;\;\;\left(\left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}\right) \cdot \frac{-1}{\left(\cos y \cdot \frac{1}{2}\right) \cdot t\_2 - \left(\left(\cos x \cdot \frac{1}{2}\right) \cdot t\_0 - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right) \cdot \cos y - t\_3}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1
        (-
         (*
          (- (cos y) (cos x))
          (* (* (sin y) (sqrt 2)) (- (sin x) (* 1/16 (sin y)))))
         2))
       (t_2 (- (sqrt 5) 3))
       (t_3 (- (* (* 1/2 t_0) (cos x)) -1)))
  (if (<= y -8358680908399641/288230376151711744)
    (/ (/ t_1 (- (* (* t_2 1/2) (cos y)) t_3)) 3)
    (if (<= y 1080863910568919/72057594037927936)
      (*
       (*
        (-
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 y)) (sqrt 2))))
         -2)
        1/3)
       (/
        -1
        (- (* (* (cos y) 1/2) t_2) (- (* (* (cos x) 1/2) t_0) -1))))
      (/
       (/ t_1 (- (* (* (- (* (sqrt 5) 1/3) 1) 3/2) (cos y)) t_3))
       3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0;
	double t_2 = sqrt(5.0) - 3.0;
	double t_3 = ((0.5 * t_0) * cos(x)) - -1.0;
	double tmp;
	if (y <= -0.029) {
		tmp = (t_1 / (((t_2 * 0.5) * cos(y)) - t_3)) / 3.0;
	} else if (y <= 0.015) {
		tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((cos(y) * 0.5) * t_2) - (((cos(x) * 0.5) * t_0) - -1.0)));
	} else {
		tmp = (t_1 / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0
    t_2 = sqrt(5.0d0) - 3.0d0
    t_3 = ((0.5d0 * t_0) * cos(x)) - (-1.0d0)
    if (y <= (-0.029d0)) then
        tmp = (t_1 / (((t_2 * 0.5d0) * cos(y)) - t_3)) / 3.0d0
    else if (y <= 0.015d0) then
        tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) * ((-1.0d0) / (((cos(y) * 0.5d0) * t_2) - (((cos(x) * 0.5d0) * t_0) - (-1.0d0))))
    else
        tmp = (t_1 / (((((sqrt(5.0d0) * 0.3333333333333333d0) - 1.0d0) * 1.5d0) * cos(y)) - t_3)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((Math.cos(y) - Math.cos(x)) * ((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0;
	double t_2 = Math.sqrt(5.0) - 3.0;
	double t_3 = ((0.5 * t_0) * Math.cos(x)) - -1.0;
	double tmp;
	if (y <= -0.029) {
		tmp = (t_1 / (((t_2 * 0.5) * Math.cos(y)) - t_3)) / 3.0;
	} else if (y <= 0.015) {
		tmp = ((((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((Math.cos(y) * 0.5) * t_2) - (((Math.cos(x) * 0.5) * t_0) - -1.0)));
	} else {
		tmp = (t_1 / (((((Math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * Math.cos(y)) - t_3)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((math.cos(y) - math.cos(x)) * ((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0
	t_2 = math.sqrt(5.0) - 3.0
	t_3 = ((0.5 * t_0) * math.cos(x)) - -1.0
	tmp = 0
	if y <= -0.029:
		tmp = (t_1 / (((t_2 * 0.5) * math.cos(y)) - t_3)) / 3.0
	elif y <= 0.015:
		tmp = ((((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((math.cos(y) * 0.5) * t_2) - (((math.cos(x) * 0.5) * t_0) - -1.0)))
	else:
		tmp = (t_1 / (((((math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * math.cos(y)) - t_3)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0)
	t_2 = Float64(sqrt(5.0) - 3.0)
	t_3 = Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0)
	tmp = 0.0
	if (y <= -0.029)
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(t_2 * 0.5) * cos(y)) - t_3)) / 3.0);
	elseif (y <= 0.015)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * Float64(-1.0 / Float64(Float64(Float64(cos(y) * 0.5) * t_2) - Float64(Float64(Float64(cos(x) * 0.5) * t_0) - -1.0))));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0;
	t_2 = sqrt(5.0) - 3.0;
	t_3 = ((0.5 * t_0) * cos(x)) - -1.0;
	tmp = 0.0;
	if (y <= -0.029)
		tmp = (t_1 / (((t_2 * 0.5) * cos(y)) - t_3)) / 3.0;
	elseif (y <= 0.015)
		tmp = ((((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) * (-1.0 / (((cos(y) * 0.5) * t_2) - (((cos(x) * 0.5) * t_0) - -1.0)));
	else
		tmp = (t_1 / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[y, -8358680908399641/288230376151711744], N[(N[(t$95$1 / N[(N[(N[(t$95$2 * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[y, 1080863910568919/72057594037927936], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] * N[(-1 / N[(N[(N[(N[Cos[y], $MachinePrecision] * 1/2), $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[(N[(N[Cos[x], $MachinePrecision] * 1/2), $MachinePrecision] * t$95$0), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] * 1/3), $MachinePrecision] - 1), $MachinePrecision] * 3/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.029000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   6. Applied rewrites 64.1%

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

   if -0.029000000000000001 < y < 0.014999999999999999

   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

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

      1. lower-*.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.5%

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

   if 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-to-mult-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. div-sub N/A

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

      15. metadata-eval N/A

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

      16. mult-flip-rev N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. sub-to-mult-rev N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   10. Applied rewrites 64.1%

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

Alternative 9: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\cos y - \cos x\right) \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2\\ t_2 := \sqrt{5} - 3\\ t_3 := \left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\\ \mathbf{if}\;y \leq \frac{-8358680908399641}{288230376151711744}:\\ \;\;\;\;\frac{\frac{t\_1}{\left(t\_2 \cdot \frac{1}{2}\right) \cdot \cos y - t\_3}}{3}\\ \mathbf{elif}\;y \leq \frac{1080863910568919}{72057594037927936}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(t\_0 \cdot \cos x - t\_2\right) \cdot \frac{1}{2} - -1\right) \cdot 3 + \left(\frac{-3}{4} \cdot \left(y \cdot y\right)\right) \cdot \left(3 - \sqrt{5}\right)}{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot y\right) \cdot \sqrt{2}\right)\right) - -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right) \cdot \cos y - t\_3}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1
        (-
         (*
          (- (cos y) (cos x))
          (* (* (sin y) (sqrt 2)) (- (sin x) (* 1/16 (sin y)))))
         2))
       (t_2 (- (sqrt 5) 3))
       (t_3 (- (* (* 1/2 t_0) (cos x)) -1)))
  (if (<= y -8358680908399641/288230376151711744)
    (/ (/ t_1 (- (* (* t_2 1/2) (cos y)) t_3)) 3)
    (if (<= y 1080863910568919/72057594037927936)
      (/
       1
       (/
        (+
         (* (- (* (- (* t_0 (cos x)) t_2) 1/2) -1) 3)
         (* (* -3/4 (* y y)) (- 3 (sqrt 5))))
        (-
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 y)) (sqrt 2))))
         -2)))
      (/
       (/ t_1 (- (* (* (- (* (sqrt 5) 1/3) 1) 3/2) (cos y)) t_3))
       3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0;
	double t_2 = sqrt(5.0) - 3.0;
	double t_3 = ((0.5 * t_0) * cos(x)) - -1.0;
	double tmp;
	if (y <= -0.029) {
		tmp = (t_1 / (((t_2 * 0.5) * cos(y)) - t_3)) / 3.0;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_0 * cos(x)) - t_2) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	} else {
		tmp = (t_1 / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0
    t_2 = sqrt(5.0d0) - 3.0d0
    t_3 = ((0.5d0 * t_0) * cos(x)) - (-1.0d0)
    if (y <= (-0.029d0)) then
        tmp = (t_1 / (((t_2 * 0.5d0) * cos(y)) - t_3)) / 3.0d0
    else if (y <= 0.015d0) then
        tmp = 1.0d0 / (((((((t_0 * cos(x)) - t_2) * 0.5d0) - (-1.0d0)) * 3.0d0) + (((-0.75d0) * (y * y)) * (3.0d0 - sqrt(5.0d0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)))) - (-2.0d0)))
    else
        tmp = (t_1 / (((((sqrt(5.0d0) * 0.3333333333333333d0) - 1.0d0) * 1.5d0) * cos(y)) - t_3)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((Math.cos(y) - Math.cos(x)) * ((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0;
	double t_2 = Math.sqrt(5.0) - 3.0;
	double t_3 = ((0.5 * t_0) * Math.cos(x)) - -1.0;
	double tmp;
	if (y <= -0.029) {
		tmp = (t_1 / (((t_2 * 0.5) * Math.cos(y)) - t_3)) / 3.0;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_0 * Math.cos(x)) - t_2) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - Math.sqrt(5.0)))) / (((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)))) - -2.0));
	} else {
		tmp = (t_1 / (((((Math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * Math.cos(y)) - t_3)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((math.cos(y) - math.cos(x)) * ((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0
	t_2 = math.sqrt(5.0) - 3.0
	t_3 = ((0.5 * t_0) * math.cos(x)) - -1.0
	tmp = 0
	if y <= -0.029:
		tmp = (t_1 / (((t_2 * 0.5) * math.cos(y)) - t_3)) / 3.0
	elif y <= 0.015:
		tmp = 1.0 / (((((((t_0 * math.cos(x)) - t_2) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - math.sqrt(5.0)))) / (((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)))) - -2.0))
	else:
		tmp = (t_1 / (((((math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * math.cos(y)) - t_3)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0)
	t_2 = Float64(sqrt(5.0) - 3.0)
	t_3 = Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0)
	tmp = 0.0
	if (y <= -0.029)
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(t_2 * 0.5) * cos(y)) - t_3)) / 3.0);
	elseif (y <= 0.015)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_0 * cos(x)) - t_2) * 0.5) - -1.0) * 3.0) + Float64(Float64(-0.75 * Float64(y * y)) * Float64(3.0 - sqrt(5.0)))) / Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)))) - -2.0)));
	else
		tmp = Float64(Float64(t_1 / Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0;
	t_2 = sqrt(5.0) - 3.0;
	t_3 = ((0.5 * t_0) * cos(x)) - -1.0;
	tmp = 0.0;
	if (y <= -0.029)
		tmp = (t_1 / (((t_2 * 0.5) * cos(y)) - t_3)) / 3.0;
	elseif (y <= 0.015)
		tmp = 1.0 / (((((((t_0 * cos(x)) - t_2) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	else
		tmp = (t_1 / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_3)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[y, -8358680908399641/288230376151711744], N[(N[(t$95$1 / N[(N[(N[(t$95$2 * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[y, 1080863910568919/72057594037927936], N[(1 / N[(N[(N[(N[(N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] * 1/2), $MachinePrecision] - -1), $MachinePrecision] * 3), $MachinePrecision] + N[(N[(-3/4 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] * 1/3), $MachinePrecision] - 1), $MachinePrecision] * 3/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.029000000000000001

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   6. Applied rewrites 64.1%

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

   if -0.029000000000000001 < y < 0.014999999999999999

   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

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

      1. lower-*.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 50.6%

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

   if 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-to-mult-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. div-sub N/A

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

      15. metadata-eval N/A

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

      16. mult-flip-rev N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. sub-to-mult-rev N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   10. Applied rewrites 64.1%

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

Alternative 10: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \sqrt{5} - 3\\ t_2 := \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(t\_1 \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\right)}}{3}\\ \mathbf{if}\;y \leq \frac{-8358680908399641}{288230376151711744}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq \frac{1080863910568919}{72057594037927936}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(t\_0 \cdot \cos x - t\_1\right) \cdot \frac{1}{2} - -1\right) \cdot 3 + \left(\frac{-3}{4} \cdot \left(y \cdot y\right)\right) \cdot \left(3 - \sqrt{5}\right)}{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot y\right) \cdot \sqrt{2}\right)\right) - -2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1 (- (sqrt 5) 3))
       (t_2
        (/
         (/
          (-
           (*
            (- (cos y) (cos x))
            (* (* (sin y) (sqrt 2)) (- (sin x) (* 1/16 (sin y)))))
           2)
          (- (* (* t_1 1/2) (cos y)) (- (* (* 1/2 t_0) (cos x)) -1)))
         3)))
  (if (<= y -8358680908399641/288230376151711744)
    t_2
    (if (<= y 1080863910568919/72057594037927936)
      (/
       1
       (/
        (+
         (* (- (* (- (* t_0 (cos x)) t_1) 1/2) -1) 3)
         (* (* -3/4 (* y y)) (- 3 (sqrt 5))))
        (-
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 y)) (sqrt 2))))
         -2)))
      t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = sqrt(5.0) - 3.0;
	double t_2 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((t_1 * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (y <= -0.029) {
		tmp = t_2;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_0 * cos(x)) - t_1) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = sqrt(5.0d0) - 3.0d0
    t_2 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / (((t_1 * 0.5d0) * cos(y)) - (((0.5d0 * t_0) * cos(x)) - (-1.0d0)))) / 3.0d0
    if (y <= (-0.029d0)) then
        tmp = t_2
    else if (y <= 0.015d0) then
        tmp = 1.0d0 / (((((((t_0 * cos(x)) - t_1) * 0.5d0) - (-1.0d0)) * 3.0d0) + (((-0.75d0) * (y * y)) * (3.0d0 - sqrt(5.0d0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)))) - (-2.0d0)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = Math.sqrt(5.0) - 3.0;
	double t_2 = ((((Math.cos(y) - Math.cos(x)) * ((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0) / (((t_1 * 0.5) * Math.cos(y)) - (((0.5 * t_0) * Math.cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (y <= -0.029) {
		tmp = t_2;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_0 * Math.cos(x)) - t_1) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - Math.sqrt(5.0)))) / (((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)))) - -2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = math.sqrt(5.0) - 3.0
	t_2 = ((((math.cos(y) - math.cos(x)) * ((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / (((t_1 * 0.5) * math.cos(y)) - (((0.5 * t_0) * math.cos(x)) - -1.0))) / 3.0
	tmp = 0
	if y <= -0.029:
		tmp = t_2
	elif y <= 0.015:
		tmp = 1.0 / (((((((t_0 * math.cos(x)) - t_1) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - math.sqrt(5.0)))) / (((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)))) - -2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(sqrt(5.0) - 3.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0) / Float64(Float64(Float64(t_1 * 0.5) * cos(y)) - Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0))) / 3.0)
	tmp = 0.0
	if (y <= -0.029)
		tmp = t_2;
	elseif (y <= 0.015)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_0 * cos(x)) - t_1) * 0.5) - -1.0) * 3.0) + Float64(Float64(-0.75 * Float64(y * y)) * Float64(3.0 - sqrt(5.0)))) / Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)))) - -2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = sqrt(5.0) - 3.0;
	t_2 = ((((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (((t_1 * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	tmp = 0.0;
	if (y <= -0.029)
		tmp = t_2;
	elseif (y <= 0.015)
		tmp = 1.0 / (((((((t_0 * cos(x)) - t_1) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(t$95$1 * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]}, If[LessEqual[y, -8358680908399641/288230376151711744], t$95$2, If[LessEqual[y, 1080863910568919/72057594037927936], N[(1 / N[(N[(N[(N[(N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * 1/2), $MachinePrecision] - -1), $MachinePrecision] * 3), $MachinePrecision] + N[(N[(-3/4 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.029000000000000001 or 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   6. Applied rewrites 64.1%

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

   if -0.029000000000000001 < y < 0.014999999999999999

   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

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

      1. lower-*.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 50.6%

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

Alternative 11: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\left(\cos y - \cos x\right) \cdot \left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{-3 \cdot \left(\left(\left(\frac{1}{2} \cdot t\_1\right) \cdot \cos x - -1\right) - \left(t\_0 \cdot \frac{1}{2}\right) \cdot \cos y\right)}\\ \mathbf{if}\;y \leq \frac{-8358680908399641}{288230376151711744}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq \frac{1080863910568919}{72057594037927936}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(t\_1 \cdot \cos x - t\_0\right) \cdot \frac{1}{2} - -1\right) \cdot 3 + \left(\frac{-3}{4} \cdot \left(y \cdot y\right)\right) \cdot \left(3 - \sqrt{5}\right)}{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot y\right) \cdot \sqrt{2}\right)\right) - -2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 3))
       (t_1 (- (sqrt 5) 1))
       (t_2
        (/
         (-
          (*
           (- (cos y) (cos x))
           (* (* (sin y) (sqrt 2)) (- (sin x) (* 1/16 (sin y)))))
          2)
         (*
          -3
          (-
           (- (* (* 1/2 t_1) (cos x)) -1)
           (* (* t_0 1/2) (cos y)))))))
  (if (<= y -8358680908399641/288230376151711744)
    t_2
    (if (<= y 1080863910568919/72057594037927936)
      (/
       1
       (/
        (+
         (* (- (* (- (* t_1 (cos x)) t_0) 1/2) -1) 3)
         (* (* -3/4 (* y y)) (- 3 (sqrt 5))))
        (-
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 y)) (sqrt 2))))
         -2)))
      t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (-3.0 * ((((0.5 * t_1) * cos(x)) - -1.0) - ((t_0 * 0.5) * cos(y))));
	double tmp;
	if (y <= -0.029) {
		tmp = t_2;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_1 * cos(x)) - t_0) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 3.0d0
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = (((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / ((-3.0d0) * ((((0.5d0 * t_1) * cos(x)) - (-1.0d0)) - ((t_0 * 0.5d0) * cos(y))))
    if (y <= (-0.029d0)) then
        tmp = t_2
    else if (y <= 0.015d0) then
        tmp = 1.0d0 / (((((((t_1 * cos(x)) - t_0) * 0.5d0) - (-1.0d0)) * 3.0d0) + (((-0.75d0) * (y * y)) * (3.0d0 - sqrt(5.0d0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)))) - (-2.0d0)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 3.0;
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = (((Math.cos(y) - Math.cos(x)) * ((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0) / (-3.0 * ((((0.5 * t_1) * Math.cos(x)) - -1.0) - ((t_0 * 0.5) * Math.cos(y))));
	double tmp;
	if (y <= -0.029) {
		tmp = t_2;
	} else if (y <= 0.015) {
		tmp = 1.0 / (((((((t_1 * Math.cos(x)) - t_0) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - Math.sqrt(5.0)))) / (((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)))) - -2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 3.0
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = (((math.cos(y) - math.cos(x)) * ((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / (-3.0 * ((((0.5 * t_1) * math.cos(x)) - -1.0) - ((t_0 * 0.5) * math.cos(y))))
	tmp = 0
	if y <= -0.029:
		tmp = t_2
	elif y <= 0.015:
		tmp = 1.0 / (((((((t_1 * math.cos(x)) - t_0) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - math.sqrt(5.0)))) / (((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)))) - -2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0) / Float64(-3.0 * Float64(Float64(Float64(Float64(0.5 * t_1) * cos(x)) - -1.0) - Float64(Float64(t_0 * 0.5) * cos(y)))))
	tmp = 0.0
	if (y <= -0.029)
		tmp = t_2;
	elseif (y <= 0.015)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(Float64(Float64(t_1 * cos(x)) - t_0) * 0.5) - -1.0) * 3.0) + Float64(Float64(-0.75 * Float64(y * y)) * Float64(3.0 - sqrt(5.0)))) / Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)))) - -2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 3.0;
	t_1 = sqrt(5.0) - 1.0;
	t_2 = (((cos(y) - cos(x)) * ((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (-3.0 * ((((0.5 * t_1) * cos(x)) - -1.0) - ((t_0 * 0.5) * cos(y))));
	tmp = 0.0;
	if (y <= -0.029)
		tmp = t_2;
	elseif (y <= 0.015)
		tmp = 1.0 / (((((((t_1 * cos(x)) - t_0) * 0.5) - -1.0) * 3.0) + ((-0.75 * (y * y)) * (3.0 - sqrt(5.0)))) / (((cos(x) - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * y)) * sqrt(2.0)))) - -2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(-3 * N[(N[(N[(N[(1/2 * t$95$1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(t$95$0 * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8358680908399641/288230376151711744], t$95$2, If[LessEqual[y, 1080863910568919/72057594037927936], N[(1 / N[(N[(N[(N[(N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] * 1/2), $MachinePrecision] - -1), $MachinePrecision] * 3), $MachinePrecision] + N[(N[(-3/4 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.029000000000000001 or 0.014999999999999999 < y

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.1%

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

   6. Applied rewrites 64.1%

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

   if -0.029000000000000001 < y < 0.014999999999999999

   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

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

      1. lower-*.f64 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 50.6%

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

Alternative 12: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_1 := t\_0 \cdot \cos x - -1\\ t_2 := \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -6600000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\left(\cos y - 1\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{t\_2 - \left(t\_0 \cdot 1 - -1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\frac{-1}{16} \cdot \left(t\_3 \cdot \sqrt{2}\right)\right) - 2}{t\_2 - t\_1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 1/2 (- (sqrt 5) 1)))
       (t_1 (- (* t_0 (cos x)) -1))
       (t_2 (* (* (- (sqrt 5) 3) 1/2) (cos y)))
       (t_3 (pow (sin x) 2)))
  (if (<= x -6600000000)
    (/
     (/
      (- (* -1/16 (* t_3 (* (sqrt 2) (- 1 (cos x))))) 2)
      (- (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y)) t_1))
     3)
    (if (<= x 1100000000000)
      (/
       (/
        (-
         (*
          (- (cos y) 1)
          (*
           (* (- (sin y) (* 1/16 (sin x))) (sqrt 2))
           (- (sin x) (* 1/16 (sin y)))))
         2)
        (- t_2 (- (* t_0 1) -1)))
       3)
      (/
       (/
        (- (* (- (cos y) (cos x)) (* -1/16 (* t_3 (sqrt 2)))) 2)
        (- t_2 t_1))
       3)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = (t_0 * cos(x)) - -1.0;
	double t_2 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_3 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = ((((cos(y) - 1.0) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (t_2 - ((t_0 * 1.0) - -1.0))) / 3.0;
	} else {
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_3 * sqrt(2.0)))) - 2.0) / (t_2 - t_1)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_1 = (t_0 * cos(x)) - (-1.0d0)
    t_2 = ((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)
    t_3 = sin(x) ** 2.0d0
    if (x <= (-6600000000.0d0)) then
        tmp = ((((-0.0625d0) * (t_3 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    else if (x <= 1100000000000.0d0) then
        tmp = ((((cos(y) - 1.0d0) * (((sin(y) - (0.0625d0 * sin(x))) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y))))) - 2.0d0) / (t_2 - ((t_0 * 1.0d0) - (-1.0d0)))) / 3.0d0
    else
        tmp = ((((cos(y) - cos(x)) * ((-0.0625d0) * (t_3 * sqrt(2.0d0)))) - 2.0d0) / (t_2 - t_1)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_1 = (t_0 * Math.cos(x)) - -1.0;
	double t_2 = ((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y);
	double t_3 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_3 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = ((((Math.cos(y) - 1.0) * (((Math.sin(y) - (0.0625 * Math.sin(x))) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y))))) - 2.0) / (t_2 - ((t_0 * 1.0) - -1.0))) / 3.0;
	} else {
		tmp = ((((Math.cos(y) - Math.cos(x)) * (-0.0625 * (t_3 * Math.sqrt(2.0)))) - 2.0) / (t_2 - t_1)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_1 = (t_0 * math.cos(x)) - -1.0
	t_2 = ((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)
	t_3 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -6600000000.0:
		tmp = (((-0.0625 * (t_3 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - t_1)) / 3.0
	elif x <= 1100000000000.0:
		tmp = ((((math.cos(y) - 1.0) * (((math.sin(y) - (0.0625 * math.sin(x))) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y))))) - 2.0) / (t_2 - ((t_0 * 1.0) - -1.0))) / 3.0
	else:
		tmp = ((((math.cos(y) - math.cos(x)) * (-0.0625 * (t_3 * math.sqrt(2.0)))) - 2.0) / (t_2 - t_1)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(Float64(t_0 * cos(x)) - -1.0)
	t_2 = Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -6600000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_3 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0);
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(y) - 1.0) * Float64(Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y))))) - 2.0) / Float64(t_2 - Float64(Float64(t_0 * 1.0) - -1.0))) / 3.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(-0.0625 * Float64(t_3 * sqrt(2.0)))) - 2.0) / Float64(t_2 - t_1)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (sqrt(5.0) - 1.0);
	t_1 = (t_0 * cos(x)) - -1.0;
	t_2 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	t_3 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -6600000000.0)
		tmp = (((-0.0625 * (t_3 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	elseif (x <= 1100000000000.0)
		tmp = ((((cos(y) - 1.0) * (((sin(y) - (0.0625 * sin(x))) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y))))) - 2.0) / (t_2 - ((t_0 * 1.0) - -1.0))) / 3.0;
	else
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_3 * sqrt(2.0)))) - 2.0) / (t_2 - t_1)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, If[LessEqual[x, -6600000000], N[(N[(N[(N[(-1/16 * N[(t$95$3 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(1/16 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(t$95$2 - N[(N[(t$95$0 * 1), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1/16 * N[(t$95$3 * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6e9

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -6.6e9 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 59.8%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 13: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -6600000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\left(\cos y - 1\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - 2}{-3 \cdot \left(\left(1 \cdot \left(t\_0 \cdot \frac{1}{2}\right) - -1\right) - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\frac{-1}{16} \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1 (- (* (* 1/2 t_0) (cos x)) -1))
       (t_2 (pow (sin x) 2)))
  (if (<= x -6600000000)
    (/
     (/
      (- (* -1/16 (* t_2 (* (sqrt 2) (- 1 (cos x))))) 2)
      (- (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y)) t_1))
     3)
    (if (<= x 1100000000000)
      (/
       (-
        (*
         (- (cos y) 1)
         (*
          (- (sin y) (* (sin x) 1/16))
          (* (- (sin x) (* 1/16 (sin y))) (sqrt 2))))
        2)
       (*
        -3
        (-
         (- (* 1 (* t_0 1/2)) -1)
         (* (* -1/2 (- 3 (sqrt 5))) (cos y)))))
      (/
       (/
        (- (* (- (cos y) (cos x)) (* -1/16 (* t_2 (sqrt 2)))) 2)
        (- (* (* (- (sqrt 5) 3) 1/2) (cos y)) t_1))
       3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = (((cos(y) - 1.0) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)))) - 2.0) / (-3.0 * (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y))));
	} else {
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((0.5d0 * t_0) * cos(x)) - (-1.0d0)
    t_2 = sin(x) ** 2.0d0
    if (x <= (-6600000000.0d0)) then
        tmp = ((((-0.0625d0) * (t_2 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    else if (x <= 1100000000000.0d0) then
        tmp = (((cos(y) - 1.0d0) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - 2.0d0) / ((-3.0d0) * (((1.0d0 * (t_0 * 0.5d0)) - (-1.0d0)) - (((-0.5d0) * (3.0d0 - sqrt(5.0d0))) * cos(y))))
    else
        tmp = ((((cos(y) - cos(x)) * ((-0.0625d0) * (t_2 * sqrt(2.0d0)))) - 2.0d0) / ((((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * Math.cos(x)) - -1.0;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_2 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = (((Math.cos(y) - 1.0) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - 2.0) / (-3.0 * (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - Math.sqrt(5.0))) * Math.cos(y))));
	} else {
		tmp = ((((Math.cos(y) - Math.cos(x)) * (-0.0625 * (t_2 * Math.sqrt(2.0)))) - 2.0) / ((((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.5 * t_0) * math.cos(x)) - -1.0
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -6600000000.0:
		tmp = (((-0.0625 * (t_2 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - t_1)) / 3.0
	elif x <= 1100000000000.0:
		tmp = (((math.cos(y) - 1.0) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - 2.0) / (-3.0 * (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - math.sqrt(5.0))) * math.cos(y))))
	else:
		tmp = ((((math.cos(y) - math.cos(x)) * (-0.0625 * (t_2 * math.sqrt(2.0)))) - 2.0) / ((((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)) - t_1)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -6600000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0);
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(cos(y) - 1.0) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - 2.0) / Float64(-3.0 * Float64(Float64(Float64(1.0 * Float64(t_0 * 0.5)) - -1.0) - Float64(Float64(-0.5 * Float64(3.0 - sqrt(5.0))) * cos(y)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(-0.0625 * Float64(t_2 * sqrt(2.0)))) - 2.0) / Float64(Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -6600000000.0)
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	elseif (x <= 1100000000000.0)
		tmp = (((cos(y) - 1.0) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)))) - 2.0) / (-3.0 * (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y))));
	else
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, If[LessEqual[x, -6600000000], N[(N[(N[(N[(-1/16 * N[(t$95$2 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(-3 * N[(N[(N[(1 * N[(t$95$0 * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1/16 * N[(t$95$2 * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6e9

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -6.6e9 < x < 1.1e12

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.8%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 14: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_1 := t\_0 \cdot \cos x - -1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -6600000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}}{\left(1 + t\_0\right) - \frac{-1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\frac{-1}{16} \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 1/2 (- (sqrt 5) 1)))
       (t_1 (- (* t_0 (cos x)) -1))
       (t_2 (pow (sin x) 2)))
  (if (<= x -6600000000)
    (/
     (/
      (- (* -1/16 (* t_2 (* (sqrt 2) (- 1 (cos x))))) 2)
      (- (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y)) t_1))
     3)
    (if (<= x 1100000000000)
      (/
       (*
        (-
         (*
          (- 1 (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- (sin x) (* 1/16 (sin y))) (sqrt 2))))
         -2)
        1/3)
       (- (+ 1 t_0) (* -1/2 (* (cos y) (- 3 (sqrt 5))))))
      (/
       (/
        (- (* (- (cos y) (cos x)) (* -1/16 (* t_2 (sqrt 2)))) 2)
        (- (* (* (- (sqrt 5) 3) 1/2) (cos y)) t_1))
       3)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = (t_0 * cos(x)) - -1.0;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / ((1.0 + t_0) - (-0.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	} else {
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_1 = (t_0 * cos(x)) - (-1.0d0)
    t_2 = sin(x) ** 2.0d0
    if (x <= (-6600000000.0d0)) then
        tmp = ((((-0.0625d0) * (t_2 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    else if (x <= 1100000000000.0d0) then
        tmp = ((((1.0d0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) / ((1.0d0 + t_0) - ((-0.5d0) * (cos(y) * (3.0d0 - sqrt(5.0d0)))))
    else
        tmp = ((((cos(y) - cos(x)) * ((-0.0625d0) * (t_2 * sqrt(2.0d0)))) - 2.0d0) / ((((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_1 = (t_0 * Math.cos(x)) - -1.0;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -6600000000.0) {
		tmp = (((-0.0625 * (t_2 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	} else if (x <= 1100000000000.0) {
		tmp = ((((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / ((1.0 + t_0) - (-0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))));
	} else {
		tmp = ((((Math.cos(y) - Math.cos(x)) * (-0.0625 * (t_2 * Math.sqrt(2.0)))) - 2.0) / ((((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_1 = (t_0 * math.cos(x)) - -1.0
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -6600000000.0:
		tmp = (((-0.0625 * (t_2 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - t_1)) / 3.0
	elif x <= 1100000000000.0:
		tmp = ((((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / ((1.0 + t_0) - (-0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))
	else:
		tmp = ((((math.cos(y) - math.cos(x)) * (-0.0625 * (t_2 * math.sqrt(2.0)))) - 2.0) / ((((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)) - t_1)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(Float64(t_0 * cos(x)) - -1.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -6600000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0);
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / Float64(Float64(1.0 + t_0) - Float64(-0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(-0.0625 * Float64(t_2 * sqrt(2.0)))) - 2.0) / Float64(Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (sqrt(5.0) - 1.0);
	t_1 = (t_0 * cos(x)) - -1.0;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -6600000000.0)
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	elseif (x <= 1100000000000.0)
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / ((1.0 + t_0) - (-0.5 * (cos(y) * (3.0 - sqrt(5.0)))));
	else
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, If[LessEqual[x, -6600000000], N[(N[(N[(N[(-1/16 * N[(t$95$2 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(N[(1 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(1 + t$95$0), $MachinePrecision] - N[(-1/2 * N[(N[Cos[y], $MachinePrecision] * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1/16 * N[(t$95$2 * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6e9

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -6.6e9 < x < 1.1e12

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.7%

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

   15. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower--.f64 N/A

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

       10. lower-sqrt.f64 59.7%

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

   17. Applied rewrites 59.7%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 15: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \mathbf{elif}\;x \leq 215000000:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}}{\left(1 \cdot \left(t\_0 \cdot \frac{1}{2}\right) - -1\right) - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\frac{-1}{16} \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1 (- (* (* 1/2 t_0) (cos x)) -1))
       (t_2 (pow (sin x) 2)))
  (if (<= x -62)
    (/
     (/
      (- (* -1/16 (* t_2 (* (sqrt 2) (- 1 (cos x))))) 2)
      (- (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y)) t_1))
     3)
    (if (<= x 215000000)
      (/
       (*
        (-
         (*
          (- 1 (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- x (* 1/16 (sin y))) (sqrt 2))))
         -2)
        1/3)
       (-
        (- (* 1 (* t_0 1/2)) -1)
        (* (* -1/2 (- 3 (sqrt 5))) (cos y))))
      (/
       (/
        (- (* (- (cos y) (cos x)) (* -1/16 (* t_2 (sqrt 2)))) 2)
        (- (* (* (- (sqrt 5) 3) 1/2) (cos y)) t_1))
       3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -62.0) {
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	} else {
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((0.5d0 * t_0) * cos(x)) - (-1.0d0)
    t_2 = sin(x) ** 2.0d0
    if (x <= (-62.0d0)) then
        tmp = ((((-0.0625d0) * (t_2 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    else if (x <= 215000000.0d0) then
        tmp = ((((1.0d0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((x - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) / (((1.0d0 * (t_0 * 0.5d0)) - (-1.0d0)) - (((-0.5d0) * (3.0d0 - sqrt(5.0d0))) * cos(y)))
    else
        tmp = ((((cos(y) - cos(x)) * ((-0.0625d0) * (t_2 * sqrt(2.0d0)))) - 2.0d0) / ((((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * Math.cos(x)) - -1.0;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -62.0) {
		tmp = (((-0.0625 * (t_2 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((x - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - Math.sqrt(5.0))) * Math.cos(y)));
	} else {
		tmp = ((((Math.cos(y) - Math.cos(x)) * (-0.0625 * (t_2 * Math.sqrt(2.0)))) - 2.0) / ((((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.5 * t_0) * math.cos(x)) - -1.0
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -62.0:
		tmp = (((-0.0625 * (t_2 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - t_1)) / 3.0
	elif x <= 215000000.0:
		tmp = ((((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((x - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - math.sqrt(5.0))) * math.cos(y)))
	else:
		tmp = ((((math.cos(y) - math.cos(x)) * (-0.0625 * (t_2 * math.sqrt(2.0)))) - 2.0) / ((((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)) - t_1)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0);
	elseif (x <= 215000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(x - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / Float64(Float64(Float64(1.0 * Float64(t_0 * 0.5)) - -1.0) - Float64(Float64(-0.5 * Float64(3.0 - sqrt(5.0))) * cos(y))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(-0.0625 * Float64(t_2 * sqrt(2.0)))) - 2.0) / Float64(Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	elseif (x <= 215000000.0)
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	else
		tmp = ((((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / ((((sqrt(5.0) - 3.0) * 0.5) * cos(y)) - t_1)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, If[LessEqual[x, -62], N[(N[(N[(N[(-1/16 * N[(t$95$2 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[x, 215000000], N[(N[(N[(N[(N[(1 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(x - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(N[(1 * N[(t$95$0 * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1/16 * N[(t$95$2 * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -62 < x < 2.15e8

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.7%

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

   15. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.2%

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

   17. Applied rewrites 55.2%

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

   if 2.15e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 16: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - t\_1}}{3}\\ \mathbf{elif}\;x \leq 215000000:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}}{\left(1 \cdot \left(t\_0 \cdot \frac{1}{2}\right) - -1\right) - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos y - \cos x\right) \cdot \left(\frac{-1}{16} \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) - 2}{-3 \cdot \left(t\_1 - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1 (- (* (* 1/2 t_0) (cos x)) -1))
       (t_2 (pow (sin x) 2)))
  (if (<= x -62)
    (/
     (/
      (- (* -1/16 (* t_2 (* (sqrt 2) (- 1 (cos x))))) 2)
      (- (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y)) t_1))
     3)
    (if (<= x 215000000)
      (/
       (*
        (-
         (*
          (- 1 (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- x (* 1/16 (sin y))) (sqrt 2))))
         -2)
        1/3)
       (-
        (- (* 1 (* t_0 1/2)) -1)
        (* (* -1/2 (- 3 (sqrt 5))) (cos y))))
      (/
       (- (* (- (cos y) (cos x)) (* -1/16 (* t_2 (sqrt 2)))) 2)
       (* -3 (- t_1 (* (* (- (sqrt 5) 3) 1/2) (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -62.0) {
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	} else {
		tmp = (((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / (-3.0 * (t_1 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((0.5d0 * t_0) * cos(x)) - (-1.0d0)
    t_2 = sin(x) ** 2.0d0
    if (x <= (-62.0d0)) then
        tmp = ((((-0.0625d0) * (t_2 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - t_1)) / 3.0d0
    else if (x <= 215000000.0d0) then
        tmp = ((((1.0d0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((x - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) / (((1.0d0 * (t_0 * 0.5d0)) - (-1.0d0)) - (((-0.5d0) * (3.0d0 - sqrt(5.0d0))) * cos(y)))
    else
        tmp = (((cos(y) - cos(x)) * ((-0.0625d0) * (t_2 * sqrt(2.0d0)))) - 2.0d0) / ((-3.0d0) * (t_1 - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = ((0.5 * t_0) * Math.cos(x)) - -1.0;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -62.0) {
		tmp = (((-0.0625 * (t_2 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - t_1)) / 3.0;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((x - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - Math.sqrt(5.0))) * Math.cos(y)));
	} else {
		tmp = (((Math.cos(y) - Math.cos(x)) * (-0.0625 * (t_2 * Math.sqrt(2.0)))) - 2.0) / (-3.0 * (t_1 - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.5 * t_0) * math.cos(x)) - -1.0
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -62.0:
		tmp = (((-0.0625 * (t_2 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - t_1)) / 3.0
	elif x <= 215000000.0:
		tmp = ((((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((x - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - math.sqrt(5.0))) * math.cos(y)))
	else:
		tmp = (((math.cos(y) - math.cos(x)) * (-0.0625 * (t_2 * math.sqrt(2.0)))) - 2.0) / (-3.0 * (t_1 - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0);
	elseif (x <= 215000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(x - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / Float64(Float64(Float64(1.0 * Float64(t_0 * 0.5)) - -1.0) - Float64(Float64(-0.5 * Float64(3.0 - sqrt(5.0))) * cos(y))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(y) - cos(x)) * Float64(-0.0625 * Float64(t_2 * sqrt(2.0)))) - 2.0) / Float64(-3.0 * Float64(t_1 - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.5 * t_0) * cos(x)) - -1.0;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = (((-0.0625 * (t_2 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - t_1)) / 3.0;
	elseif (x <= 215000000.0)
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	else
		tmp = (((cos(y) - cos(x)) * (-0.0625 * (t_2 * sqrt(2.0)))) - 2.0) / (-3.0 * (t_1 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, If[LessEqual[x, -62], N[(N[(N[(N[(-1/16 * N[(t$95$2 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], If[LessEqual[x, 215000000], N[(N[(N[(N[(N[(1 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(x - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(N[(1 * N[(t$95$0 * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1/16 * N[(t$95$2 * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(-3 * N[(t$95$1 - N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -62 < x < 2.15e8

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.7%

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

   15. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.2%

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

   17. Applied rewrites 55.2%

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

   if 2.15e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 17: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot t\_0\right) \cdot \cos x - -1\right)}}{3}\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 215000000:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}}{\left(1 \cdot \left(t\_0 \cdot \frac{1}{2}\right) - -1\right) - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1
        (/
         (/
          (-
           (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x)))))
           2)
          (-
           (* (* (* (- 1 (/ 3 (sqrt 5))) (sqrt 5)) 1/2) (cos y))
           (- (* (* 1/2 t_0) (cos x)) -1)))
         3)))
  (if (<= x -62)
    t_1
    (if (<= x 215000000)
      (/
       (*
        (-
         (*
          (- 1 (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- x (* 1/16 (sin y))) (sqrt 2))))
         -2)
        1/3)
       (-
        (- (* 1 (* t_0 1/2)) -1)
        (* (* -1/2 (- 3 (sqrt 5))) (cos y))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = (((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (x <= -62.0) {
		tmp = t_1;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = ((((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y)) - (((0.5d0 * t_0) * cos(x)) - (-1.0d0)))) / 3.0d0
    if (x <= (-62.0d0)) then
        tmp = t_1
    else if (x <= 215000000.0d0) then
        tmp = ((((1.0d0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((x - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) / (((1.0d0 * (t_0 * 0.5d0)) - (-1.0d0)) - (((-0.5d0) * (3.0d0 - sqrt(5.0d0))) * cos(y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = (((-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y)) - (((0.5 * t_0) * Math.cos(x)) - -1.0))) / 3.0;
	double tmp;
	if (x <= -62.0) {
		tmp = t_1;
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((x - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - Math.sqrt(5.0))) * Math.cos(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = (((-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y)) - (((0.5 * t_0) * math.cos(x)) - -1.0))) / 3.0
	tmp = 0
	if x <= -62.0:
		tmp = t_1
	elif x <= 215000000.0:
		tmp = ((((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((x - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - math.sqrt(5.0))) * math.cos(y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - Float64(Float64(Float64(0.5 * t_0) * cos(x)) - -1.0))) / 3.0)
	tmp = 0.0
	if (x <= -62.0)
		tmp = t_1;
	elseif (x <= 215000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(x - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / Float64(Float64(Float64(1.0 * Float64(t_0 * 0.5)) - -1.0) - Float64(Float64(-0.5 * Float64(3.0 - sqrt(5.0))) * cos(y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = (((-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y)) - (((0.5 * t_0) * cos(x)) - -1.0))) / 3.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = t_1;
	elseif (x <= 215000000.0)
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_0 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(1 - N[(3 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1/2 * t$95$0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]}, If[LessEqual[x, -62], t$95$1, If[LessEqual[x, 215000000], N[(N[(N[(N[(N[(1 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(x - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(N[(1 * N[(t$95$0 * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -62 or 2.15e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   8. Applied rewrites 61.8%

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

   if -62 < x < 2.15e8

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.7%

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

   15. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.2%

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

   17. Applied rewrites 55.2%

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

Alternative 18: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\frac{1}{2} \cdot t\_1\right) \cdot \cos x - -1\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{1}{\frac{t\_2 - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}\\ \mathbf{elif}\;x \leq 215000000:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \left(\left(\sin y - \sin x \cdot \frac{1}{16}\right) \cdot \left(\left(x - \frac{1}{16} \cdot \sin y\right) \cdot \sqrt{2}\right)\right) - -2\right) \cdot \frac{1}{3}}{\left(1 \cdot \left(t\_1 \cdot \frac{1}{2}\right) - -1\right) - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right) \cdot \cos y - t\_2}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (pow (sin x) 2))
       (t_1 (- (sqrt 5) 1))
       (t_2 (- (* (* 1/2 t_1) (cos x)) -1)))
  (if (<= x -62)
    (/
     1
     (/
      (- t_2 (* (* (- (sqrt 5) 3) 1/2) (cos y)))
      (* 1/3 (+ 2 (* -1/16 (* t_0 (* (sqrt 2) (- (cos x) 1))))))))
    (if (<= x 215000000)
      (/
       (*
        (-
         (*
          (- 1 (cos y))
          (*
           (- (sin y) (* (sin x) 1/16))
           (* (- x (* 1/16 (sin y))) (sqrt 2))))
         -2)
        1/3)
       (-
        (- (* 1 (* t_1 1/2)) -1)
        (* (* -1/2 (- 3 (sqrt 5))) (cos y))))
      (/
       (/
        (- (* -1/16 (* t_0 (* (sqrt 2) (- 1 (cos x))))) 2)
        (- (* (* (- (* (sqrt 5) 1/3) 1) 3/2) (cos y)) t_2))
       3)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = ((0.5 * t_1) * cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_1 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	} else {
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = ((0.5d0 * t_1) * cos(x)) - (-1.0d0)
    if (x <= (-62.0d0)) then
        tmp = 1.0d0 / ((t_2 - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) / (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (cos(x) - 1.0d0)))))))
    else if (x <= 215000000.0d0) then
        tmp = ((((1.0d0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625d0)) * ((x - (0.0625d0 * sin(y))) * sqrt(2.0d0)))) - (-2.0d0)) * 0.3333333333333333d0) / (((1.0d0 * (t_1 * 0.5d0)) - (-1.0d0)) - (((-0.5d0) * (3.0d0 - sqrt(5.0d0))) * cos(y)))
    else
        tmp = ((((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((sqrt(5.0d0) * 0.3333333333333333d0) - 1.0d0) * 1.5d0) * cos(y)) - t_2)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = ((0.5 * t_1) * Math.cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_2 - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * (Math.cos(x) - 1.0)))))));
	} else if (x <= 215000000.0) {
		tmp = ((((1.0 - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) * 0.0625)) * ((x - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_1 * 0.5)) - -1.0) - ((-0.5 * (3.0 - Math.sqrt(5.0))) * Math.cos(y)));
	} else {
		tmp = (((-0.0625 * (t_0 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((Math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * Math.cos(y)) - t_2)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = ((0.5 * t_1) * math.cos(x)) - -1.0
	tmp = 0
	if x <= -62.0:
		tmp = 1.0 / ((t_2 - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * (math.cos(x) - 1.0)))))))
	elif x <= 215000000.0:
		tmp = ((((1.0 - math.cos(y)) * ((math.sin(y) - (math.sin(x) * 0.0625)) * ((x - (0.0625 * math.sin(y))) * math.sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_1 * 0.5)) - -1.0) - ((-0.5 * (3.0 - math.sqrt(5.0))) * math.cos(y)))
	else:
		tmp = (((-0.0625 * (t_0 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * math.cos(y)) - t_2)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(Float64(0.5 * t_1) * cos(x)) - -1.0)
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(1.0 / Float64(Float64(t_2 - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) / Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))))));
	elseif (x <= 215000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) * 0.0625)) * Float64(Float64(x - Float64(0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / Float64(Float64(Float64(1.0 * Float64(t_1 * 0.5)) - -1.0) - Float64(Float64(-0.5 * Float64(3.0 - sqrt(5.0))) * cos(y))));
	else
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = sqrt(5.0) - 1.0;
	t_2 = ((0.5 * t_1) * cos(x)) - -1.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = 1.0 / ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	elseif (x <= 215000000.0)
		tmp = ((((1.0 - cos(y)) * ((sin(y) - (sin(x) * 0.0625)) * ((x - (0.0625 * sin(y))) * sqrt(2.0)))) - -2.0) * 0.3333333333333333) / (((1.0 * (t_1 * 0.5)) - -1.0) - ((-0.5 * (3.0 - sqrt(5.0))) * cos(y)));
	else
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1/2 * t$95$1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[x, -62], N[(1 / N[(N[(t$95$2 - N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1/3 * N[(2 + N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 215000000], N[(N[(N[(N[(N[(1 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 1/16), $MachinePrecision]), $MachinePrecision] * N[(N[(x - N[(1/16 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(N[(1 * N[(t$95$1 * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(-1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] * 1/3), $MachinePrecision] - 1), $MachinePrecision] * 3/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

   if -62 < x < 2.15e8

   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

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

      2. add-flip N/A

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

      3. sub-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 60.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 59.4%

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

   14. Applied rewrites 59.7%

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

   15. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 55.2%

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

   17. Applied rewrites 55.2%

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

   if 2.15e8 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-to-mult-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. div-sub N/A

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

      15. metadata-eval N/A

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

      16. mult-flip-rev N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. sub-to-mult-rev N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   10. Applied rewrites 61.8%

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

Alternative 19: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_2 := t\_1 \cdot \cos x - -1\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{1}{\frac{t\_2 - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right) - \left(1 + t\_1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right) \cdot \cos y - t\_2}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (pow (sin x) 2))
       (t_1 (* 1/2 (- (sqrt 5) 1)))
       (t_2 (- (* t_1 (cos x)) -1)))
  (if (<= x -62)
    (/
     1
     (/
      (- t_2 (* (* (- (sqrt 5) 3) 1/2) (cos y)))
      (* 1/3 (+ 2 (* -1/16 (* t_0 (* (sqrt 2) (- (cos x) 1))))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 1/2 (* (cos y) (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5)))))))
         (+ 1 t_1)))
       3)
      (/
       (/
        (- (* -1/16 (* t_0 (* (sqrt 2) (- 1 (cos x))))) 2)
        (- (* (* (- (* (sqrt 5) 1/3) 1) 3/2) (cos y)) t_2))
       3)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 0.5 * (sqrt(5.0) - 1.0);
	double t_2 = (t_1 * cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	} else {
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_2 = (t_1 * cos(x)) - (-1.0d0)
    if (x <= (-62.0d0)) then
        tmp = 1.0d0 / ((t_2 - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) / (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (cos(x) - 1.0d0)))))))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))) - (1.0d0 + t_1))) / 3.0d0
    else
        tmp = ((((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (((((sqrt(5.0d0) * 0.3333333333333333d0) - 1.0d0) * 1.5d0) * cos(y)) - t_2)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_2 = (t_1 * Math.cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_2 - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * (Math.cos(x) - 1.0)))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	} else {
		tmp = (((-0.0625 * (t_0 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (((((Math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * Math.cos(y)) - t_2)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_2 = (t_1 * math.cos(x)) - -1.0
	tmp = 0
	if x <= -62.0:
		tmp = 1.0 / ((t_2 - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * (math.cos(x) - 1.0)))))))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))) - (1.0 + t_1))) / 3.0
	else:
		tmp = (((-0.0625 * (t_0 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (((((math.sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * math.cos(y)) - t_2)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_2 = Float64(Float64(t_1 * cos(x)) - -1.0)
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(1.0 / Float64(Float64(t_2 - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) / Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))))));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0))))))) - Float64(1.0 + t_1))) / 3.0);
	else
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(Float64(Float64(Float64(sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = 0.5 * (sqrt(5.0) - 1.0);
	t_2 = (t_1 * cos(x)) - -1.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = 1.0 / ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	else
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (((((sqrt(5.0) * 0.3333333333333333) - 1.0) * 1.5) * cos(y)) - t_2)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, Block[{t$95$1 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[x, -62], N[(1 / N[(N[(t$95$2 - N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1/3 * N[(2 + N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(N[(N[(N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(N[(N[(N[(N[Sqrt[5], $MachinePrecision] * 1/3), $MachinePrecision] - 1), $MachinePrecision] * 3/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

   if -62 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-to-mult-rev N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. distribute-neg-frac N/A

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

      11. lift--.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. div-sub N/A

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

      15. metadata-eval N/A

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

      16. mult-flip-rev N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f64 N/A

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

      19. sub-to-mult-rev N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   10. Applied rewrites 61.8%

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

Alternative 20: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\\ t_2 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_3 := t\_2 \cdot \cos x - -1\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{1}{\frac{t\_3 - t\_1}{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right) - \left(1 + t\_2\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{t\_1 - t\_3}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (pow (sin x) 2))
       (t_1 (* (* (- (sqrt 5) 3) 1/2) (cos y)))
       (t_2 (* 1/2 (- (sqrt 5) 1)))
       (t_3 (- (* t_2 (cos x)) -1)))
  (if (<= x -62)
    (/
     1
     (/
      (- t_3 t_1)
      (* 1/3 (+ 2 (* -1/16 (* t_0 (* (sqrt 2) (- (cos x) 1))))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 1/2 (* (cos y) (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5)))))))
         (+ 1 t_2)))
       3)
      (/
       (/
        (- (* -1/16 (* t_0 (* (sqrt 2) (- 1 (cos x))))) 2)
        (- t_1 t_3))
       3)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	double t_2 = 0.5 * (sqrt(5.0) - 1.0);
	double t_3 = (t_2 * cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_3 - t_1) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_2))) / 3.0;
	} else {
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (t_1 - t_3)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = ((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)
    t_2 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_3 = (t_2 * cos(x)) - (-1.0d0)
    if (x <= (-62.0d0)) then
        tmp = 1.0d0 / ((t_3 - t_1) / (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (cos(x) - 1.0d0)))))))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))) - (1.0d0 + t_2))) / 3.0d0
    else
        tmp = ((((-0.0625d0) * (t_0 * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / (t_1 - t_3)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = ((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y);
	double t_2 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_3 = (t_2 * Math.cos(x)) - -1.0;
	double tmp;
	if (x <= -62.0) {
		tmp = 1.0 / ((t_3 - t_1) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * (Math.cos(x) - 1.0)))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))) - (1.0 + t_2))) / 3.0;
	} else {
		tmp = (((-0.0625 * (t_0 * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (t_1 - t_3)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = ((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)
	t_2 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_3 = (t_2 * math.cos(x)) - -1.0
	tmp = 0
	if x <= -62.0:
		tmp = 1.0 / ((t_3 - t_1) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * (math.cos(x) - 1.0)))))))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))) - (1.0 + t_2))) / 3.0
	else:
		tmp = (((-0.0625 * (t_0 * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (t_1 - t_3)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))
	t_2 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_3 = Float64(Float64(t_2 * cos(x)) - -1.0)
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(1.0 / Float64(Float64(t_3 - t_1) / Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))))));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0))))))) - Float64(1.0 + t_2))) / 3.0);
	else
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(t_1 - t_3)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	t_2 = 0.5 * (sqrt(5.0) - 1.0);
	t_3 = (t_2 * cos(x)) - -1.0;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = 1.0 / ((t_3 - t_1) / (0.3333333333333333 * (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * (cos(x) - 1.0)))))));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_2))) / 3.0;
	else
		tmp = (((-0.0625 * (t_0 * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (t_1 - t_3)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[x, -62], N[(1 / N[(N[(t$95$3 - t$95$1), $MachinePrecision] / N[(1/3 * N[(2 + N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(N[(N[(N[(-1/16 * N[(t$95$0 * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

   if -62 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 21: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2\\ t_1 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_2 := t\_1 \cdot \cos x - -1\\ t_3 := \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;\frac{t\_0}{-3 \cdot \left(t\_2 - t\_3\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right) - \left(1 + t\_1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_3 - t\_2}}{3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (-
         (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x)))))
         2))
       (t_1 (* 1/2 (- (sqrt 5) 1)))
       (t_2 (- (* t_1 (cos x)) -1))
       (t_3 (* (* (- (sqrt 5) 3) 1/2) (cos y))))
  (if (<= x -62)
    (/ t_0 (* -3 (- t_2 t_3)))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 1/2 (* (cos y) (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5)))))))
         (+ 1 t_1)))
       3)
      (/ (/ t_0 (- t_3 t_2)) 3)))))

C

double code(double x, double y) {
	double t_0 = (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0;
	double t_1 = 0.5 * (sqrt(5.0) - 1.0);
	double t_2 = (t_1 * cos(x)) - -1.0;
	double t_3 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	double tmp;
	if (x <= -62.0) {
		tmp = t_0 / (-3.0 * (t_2 - t_3));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	} else {
		tmp = (t_0 / (t_3 - t_2)) / 3.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0
    t_1 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_2 = (t_1 * cos(x)) - (-1.0d0)
    t_3 = ((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)
    if (x <= (-62.0d0)) then
        tmp = t_0 / ((-3.0d0) * (t_2 - t_3))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))) - (1.0d0 + t_1))) / 3.0d0
    else
        tmp = (t_0 / (t_3 - t_2)) / 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0;
	double t_1 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_2 = (t_1 * Math.cos(x)) - -1.0;
	double t_3 = ((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y);
	double tmp;
	if (x <= -62.0) {
		tmp = t_0 / (-3.0 * (t_2 - t_3));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	} else {
		tmp = (t_0 / (t_3 - t_2)) / 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0
	t_1 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_2 = (t_1 * math.cos(x)) - -1.0
	t_3 = ((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)
	tmp = 0
	if x <= -62.0:
		tmp = t_0 / (-3.0 * (t_2 - t_3))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))) - (1.0 + t_1))) / 3.0
	else:
		tmp = (t_0 / (t_3 - t_2)) / 3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0)
	t_1 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_2 = Float64(Float64(t_1 * cos(x)) - -1.0)
	t_3 = Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(t_0 / Float64(-3.0 * Float64(t_2 - t_3)));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0))))))) - Float64(1.0 + t_1))) / 3.0);
	else
		tmp = Float64(Float64(t_0 / Float64(t_3 - t_2)) / 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0;
	t_1 = 0.5 * (sqrt(5.0) - 1.0);
	t_2 = (t_1 * cos(x)) - -1.0;
	t_3 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	tmp = 0.0;
	if (x <= -62.0)
		tmp = t_0 / (-3.0 * (t_2 - t_3));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_1))) / 3.0;
	else
		tmp = (t_0 / (t_3 - t_2)) / 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]}, Block[{t$95$1 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -62], N[(t$95$0 / N[(-3 * N[(t$95$2 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(N[(t$95$0 / N[(t$95$3 - t$95$2), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

   if -62 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

Alternative 22: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{-3 \cdot \left(\left(t\_0 \cdot \cos x - -1\right) - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right)}\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right) - \left(1 + t\_0\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 1/2 (- (sqrt 5) 1)))
       (t_1
        (/
         (-
          (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x)))))
          2)
         (*
          -3
          (-
           (- (* t_0 (cos x)) -1)
           (* (* (- (sqrt 5) 3) 1/2) (cos y)))))))
  (if (<= x -62)
    t_1
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 1/2 (* (cos y) (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5)))))))
         (+ 1 t_0)))
       3)
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = ((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (-3.0 * (((t_0 * cos(x)) - -1.0) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))));
	double tmp;
	if (x <= -62.0) {
		tmp = t_1;
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_0))) / 3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_1 = (((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / ((-3.0d0) * (((t_0 * cos(x)) - (-1.0d0)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))))
    if (x <= (-62.0d0)) then
        tmp = t_1
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))) - (1.0d0 + t_0))) / 3.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_1 = ((-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / (-3.0 * (((t_0 * Math.cos(x)) - -1.0) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))));
	double tmp;
	if (x <= -62.0) {
		tmp = t_1;
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))) - (1.0 + t_0))) / 3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_1 = ((-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / (-3.0 * (((t_0 * math.cos(x)) - -1.0) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))))
	tmp = 0
	if x <= -62.0:
		tmp = t_1
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))) - (1.0 + t_0))) / 3.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(-3.0 * Float64(Float64(Float64(t_0 * cos(x)) - -1.0) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y)))))
	tmp = 0.0
	if (x <= -62.0)
		tmp = t_1;
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0))))))) - Float64(1.0 + t_0))) / 3.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (sqrt(5.0) - 1.0);
	t_1 = ((-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / (-3.0 * (((t_0 * cos(x)) - -1.0) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))));
	tmp = 0.0;
	if (x <= -62.0)
		tmp = t_1;
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))) - (1.0 + t_0))) / 3.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(-3 * N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision] - N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision] * 1/2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -62], t$95$1, If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -62 or 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.8%

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

   6. Applied rewrites 61.8%

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

   if -62 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

Alternative 23: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\\ t_2 := \frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + t\_0 \cdot \cos x\right) - -1\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \left(\frac{2}{t\_2} - \frac{\left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}{t\_2}\right)\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot t\_1\right) - \left(1 + \frac{1}{2} \cdot t\_0\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot t\_1 - \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot t\_0\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1))
       (t_1 (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5))))))
       (t_2 (- (* 1/2 (+ (- 3 (sqrt 5)) (* t_0 (cos x)))) -1)))
  (if (<= x -1320)
    (*
     1/3
     (-
      (/ 2 t_2)
      (/
       (*
        (* 1/16 (* (- (cos x) 1) (sqrt 2)))
        (- 1/2 (* 1/2 (cos (* 2 x)))))
       t_2)))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (- (* 1/2 (* (cos y) t_1)) (+ 1 (* 1/2 t_0))))
       3)
      (*
       1/3
       (/
        (- (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x))))) 2)
        (- (* 1/2 t_1) (+ 1 (* 1/2 (* (cos x) t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))));
	double t_2 = (0.5 * ((3.0 - sqrt(5.0)) + (t_0 * cos(x)))) - -1.0;
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) / t_2));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * t_1)) - (1.0 + (0.5 * t_0)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * t_1) - (1.0 + (0.5 * (cos(x) * t_0)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))
    t_2 = (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (t_0 * cos(x)))) - (-1.0d0)
    if (x <= (-1320.0d0)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 / t_2) - (((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) / t_2))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * t_1)) - (1.0d0 + (0.5d0 * t_0)))) / 3.0d0
    else
        tmp = 0.3333333333333333d0 * ((((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / ((0.5d0 * t_1) - (1.0d0 + (0.5d0 * (cos(x) * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))));
	double t_2 = (0.5 * ((3.0 - Math.sqrt(5.0)) + (t_0 * Math.cos(x)))) - -1.0;
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) / t_2));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * t_1)) - (1.0 + (0.5 * t_0)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / ((0.5 * t_1) - (1.0 + (0.5 * (Math.cos(x) * t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))
	t_2 = (0.5 * ((3.0 - math.sqrt(5.0)) + (t_0 * math.cos(x)))) - -1.0
	tmp = 0
	if x <= -1320.0:
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) / t_2))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * t_1)) - (1.0 + (0.5 * t_0)))) / 3.0
	else:
		tmp = 0.3333333333333333 * (((-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / ((0.5 * t_1) - (1.0 + (0.5 * (math.cos(x) * t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))
	t_2 = Float64(Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(t_0 * cos(x)))) - -1.0)
	tmp = 0.0
	if (x <= -1320.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 / t_2) - Float64(Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) / t_2)));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * t_1)) - Float64(1.0 + Float64(0.5 * t_0)))) / 3.0);
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(0.5 * t_1) - Float64(1.0 + Float64(0.5 * Float64(cos(x) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))));
	t_2 = (0.5 * ((3.0 - sqrt(5.0)) + (t_0 * cos(x)))) - -1.0;
	tmp = 0.0;
	if (x <= -1320.0)
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) / t_2));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * t_1)) - (1.0 + (0.5 * t_0)))) / 3.0;
	else
		tmp = 0.3333333333333333 * (((-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * t_1) - (1.0 + (0.5 * (cos(x) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1/2 * N[(N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]}, If[LessEqual[x, -1320], N[(1/3 * N[(N[(2 / t$95$2), $MachinePrecision] - N[(N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(1 + N[(1/2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(1/3 * N[(N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * t$95$1), $MachinePrecision] - N[(1 + N[(1/2 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.6%

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

   6. Applied rewrites 59.5%

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

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 59.5%

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

Alternative 24: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{1}{2} \cdot \left(\cos x \cdot t\_1\right)\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{1 + \left(t\_2 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot t\_0\right) - \left(1 + \frac{1}{2} \cdot t\_1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot t\_0 - \left(1 + t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5))))))
       (t_1 (- (sqrt 5) 1))
       (t_2 (* 1/2 (* (cos x) t_1))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+
       1
       (-
        1
        (*
         (* 1/16 (* (- (cos x) 1) (sqrt 2)))
         (- 1/2 (* 1/2 (cos (* 2 x)))))))
      (+ 1 (+ t_2 (* 1/2 (- 3 (sqrt 5)))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (- (* 1/2 (* (cos y) t_0)) (+ 1 (* 1/2 t_1))))
       3)
      (*
       1/3
       (/
        (- (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x))))) 2)
        (- (* 1/2 t_0) (+ 1 t_2))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))));
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (cos(x) * t_1);
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + (t_2 + (0.5 * (3.0 - sqrt(5.0))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * t_0)) - (1.0 + (0.5 * t_1)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * t_0) - (1.0 + t_2)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = 0.5d0 * (cos(x) * t_1)
    if (x <= (-1320.0d0)) then
        tmp = 0.3333333333333333d0 * ((1.0d0 + (1.0d0 - ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))))) / (1.0d0 + (t_2 + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((0.5d0 * (cos(y) * t_0)) - (1.0d0 + (0.5d0 * t_1)))) / 3.0d0
    else
        tmp = 0.3333333333333333d0 * ((((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / ((0.5d0 * t_0) - (1.0d0 + t_2)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))));
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = 0.5 * (Math.cos(x) * t_1);
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))))) / (1.0 + (t_2 + (0.5 * (3.0 - Math.sqrt(5.0))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (Math.cos(y) * t_0)) - (1.0 + (0.5 * t_1)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / ((0.5 * t_0) - (1.0 + t_2)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = 0.5 * (math.cos(x) * t_1)
	tmp = 0
	if x <= -1320.0:
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))))) / (1.0 + (t_2 + (0.5 * (3.0 - math.sqrt(5.0))))))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((0.5 * (math.cos(y) * t_0)) - (1.0 + (0.5 * t_1)))) / 3.0
	else:
		tmp = 0.3333333333333333 * (((-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / ((0.5 * t_0) - (1.0 + t_2)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 * Float64(cos(x) * t_1))
	tmp = 0.0
	if (x <= -1320.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(1.0 + Float64(1.0 - Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))))) / Float64(1.0 + Float64(t_2 + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(0.5 * Float64(cos(y) * t_0)) - Float64(1.0 + Float64(0.5 * t_1)))) / 3.0);
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(0.5 * t_0) - Float64(1.0 + t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))));
	t_1 = sqrt(5.0) - 1.0;
	t_2 = 0.5 * (cos(x) * t_1);
	tmp = 0.0;
	if (x <= -1320.0)
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + (t_2 + (0.5 * (3.0 - sqrt(5.0))))));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((0.5 * (cos(y) * t_0)) - (1.0 + (0.5 * t_1)))) / 3.0;
	else
		tmp = 0.3333333333333333 * (((-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * t_0) - (1.0 + t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(1/2 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1320], N[(1/3 * N[(N[(1 + N[(1 - N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1 + N[(t$95$2 + N[(1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(1 + N[(1/2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(1/3 * N[(N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * t$95$0), $MachinePrecision] - N[(1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 59.5%

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

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 59.4%

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

   if 1.1e12 < x

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   8. Applied rewrites 59.5%

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

Alternative 25: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{1}{2} \cdot \left(\cos x \cdot t\_0\right)\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{1 + \left(t\_1 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot t\_0\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right) - \left(1 + t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5) 1)) (t_1 (* 1/2 (* (cos x) t_0))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+
       1
       (-
        1
        (*
         (* 1/16 (* (- (cos x) 1) (sqrt 2)))
         (- 1/2 (* 1/2 (cos (* 2 x)))))))
      (+ 1 (+ t_1 (* 1/2 (- 3 (sqrt 5)))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 3/2 (* (cos y) (- (* 1/3 (sqrt 5)) 1)))
         (+ 1 (* 1/2 t_0))))
       3)
      (*
       1/3
       (/
        (- (* -1/16 (* (pow (sin x) 2) (* (sqrt 2) (- 1 (cos x))))) 2)
        (-
         (* 1/2 (* (sqrt 5) (- 1 (* 3 (/ 1 (sqrt 5))))))
         (+ 1 t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.5 * (cos(x) * t_0);
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + (t_1 + (0.5 * (3.0 - sqrt(5.0))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((1.5 * (cos(y) * ((0.3333333333333333 * sqrt(5.0)) - 1.0))) - (1.0 + (0.5 * t_0)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0)))))) - (1.0 + t_1)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 1.0d0
    t_1 = 0.5d0 * (cos(x) * t_0)
    if (x <= (-1320.0d0)) then
        tmp = 0.3333333333333333d0 * ((1.0d0 + (1.0d0 - ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))))) / (1.0d0 + (t_1 + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    else if (x <= 1100000000000.0d0) then
        tmp = ((((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (cos(y) - 1.0d0)))) - 2.0d0) / ((1.5d0 * (cos(y) * ((0.3333333333333333d0 * sqrt(5.0d0)) - 1.0d0))) - (1.0d0 + (0.5d0 * t_0)))) / 3.0d0
    else
        tmp = 0.3333333333333333d0 * ((((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(x))))) - 2.0d0) / ((0.5d0 * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0)))))) - (1.0d0 + t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) - 1.0;
	double t_1 = 0.5 * (Math.cos(x) * t_0);
	double tmp;
	if (x <= -1320.0) {
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))))) / (1.0 + (t_1 + (0.5 * (3.0 - Math.sqrt(5.0))))));
	} else if (x <= 1100000000000.0) {
		tmp = (((-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (Math.cos(y) - 1.0)))) - 2.0) / ((1.5 * (Math.cos(y) * ((0.3333333333333333 * Math.sqrt(5.0)) - 1.0))) - (1.0 + (0.5 * t_0)))) / 3.0;
	} else {
		tmp = 0.3333333333333333 * (((-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(x))))) - 2.0) / ((0.5 * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0)))))) - (1.0 + t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = 0.5 * (math.cos(x) * t_0)
	tmp = 0
	if x <= -1320.0:
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))))) / (1.0 + (t_1 + (0.5 * (3.0 - math.sqrt(5.0))))))
	elif x <= 1100000000000.0:
		tmp = (((-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (math.cos(y) - 1.0)))) - 2.0) / ((1.5 * (math.cos(y) * ((0.3333333333333333 * math.sqrt(5.0)) - 1.0))) - (1.0 + (0.5 * t_0)))) / 3.0
	else:
		tmp = 0.3333333333333333 * (((-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(x))))) - 2.0) / ((0.5 * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0)))))) - (1.0 + t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.5 * Float64(cos(x) * t_0))
	tmp = 0.0
	if (x <= -1320.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(1.0 + Float64(1.0 - Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))))) / Float64(1.0 + Float64(t_1 + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	elseif (x <= 1100000000000.0)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(y) - 1.0)))) - 2.0) / Float64(Float64(1.5 * Float64(cos(y) * Float64(Float64(0.3333333333333333 * sqrt(5.0)) - 1.0))) - Float64(1.0 + Float64(0.5 * t_0)))) / 3.0);
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(x))))) - 2.0) / Float64(Float64(0.5 * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))) - Float64(1.0 + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = 0.5 * (cos(x) * t_0);
	tmp = 0.0;
	if (x <= -1320.0)
		tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + (t_1 + (0.5 * (3.0 - sqrt(5.0))))));
	elseif (x <= 1100000000000.0)
		tmp = (((-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (cos(y) - 1.0)))) - 2.0) / ((1.5 * (cos(y) * ((0.3333333333333333 * sqrt(5.0)) - 1.0))) - (1.0 + (0.5 * t_0)))) / 3.0;
	else
		tmp = 0.3333333333333333 * (((-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(x))))) - 2.0) / ((0.5 * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0)))))) - (1.0 + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(1/2 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1320], N[(1/3 * N[(N[(1 + N[(1 - N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1 + N[(t$95$1 + N[(1/2 * N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1100000000000], N[(N[(N[(N[(-1/16 * N[(N[Power[N[Sin[y], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(3/2 * N[(N[Cos[y], $MachinePrecision] * N[(N[(1/3 * N[Sqrt[5], $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + N[(1/2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3), $MachinePrecision], N[(1/3 * N[(N[(N[(-1/16 * N[(N[Power[N[Sin[x], $MachinePrecision], 2], $MachinePrecision] * N[(N[Sqrt[2], $MachinePrecision] * N[(1 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(1/2 * N[(N[Sqrt[5], $MachinePrecision] * N[(1 - N[(3 * N[(1 / N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 59.6%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 59.5%

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

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sub-to-mult-rev N/A

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

      8. sub-negate-rev N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\color{blue}{\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\mathsf{neg}\left(\color{blue}{\left(3 - \sqrt{5}\right)}\right)}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      10. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right)} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{\color{blue}{3 - \sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{3 - \color{blue}{\sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{3 - \color{blue}{\sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      14. div-sub N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      16. mult-flip-rev N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      19. sub-to-mult-rev N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right)} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   10. Applied rewrites 59.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   if 1.1e12 < x

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   8. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos x\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot t\_2\right) \cdot t\_0\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot t\_1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot t\_1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot t\_0\right) \cdot t\_2 - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1/2 (* 1/2 (cos (* 2 x)))))
       (t_1 (- (sqrt 5) 1))
       (t_2 (* (- (cos x) 1) (sqrt 2))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+ 1 (- 1 (* (* 1/16 t_2) t_0)))
      (+ 1 (+ (* 1/2 (* (cos x) t_1)) (* 1/2 (- 3 (sqrt 5)))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (-
         (* 3/2 (* (cos y) (- (* 1/3 (sqrt 5)) 1)))
         (+ 1 (* 1/2 t_1))))
       3)
      (*
       1/3
       (/
        (- (* (* -1/16 t_0) t_2) -2)
        (+
         1
         (134-z0z1z2z3z4
          1/2
          t_1
          (cos x)
          (- 1 (/ 3 (sqrt 5)))
          (sqrt 5)))))))))

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(1 + 1\right) - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate--l+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      3. mult-flip-rev N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\frac{\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}}{2}} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      7. sub-to-mult-rev N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\color{blue}{\sqrt{5} - 3}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      8. sub-negate-rev N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\color{blue}{\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)}}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\frac{\mathsf{neg}\left(\color{blue}{\left(3 - \sqrt{5}\right)}\right)}{2} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      10. distribute-neg-frac N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right)} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{\color{blue}{3 - \sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{3 - \color{blue}{\sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\frac{3 - \color{blue}{\sqrt{5}}}{2}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      14. div-sub N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      16. mult-flip-rev N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

      19. sub-to-mult-rev N/A

          \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\mathsf{neg}\left(\color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{3} - 1\right) \cdot \frac{3}{2}\right)} \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   10. Applied rewrites 59.4%

       \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(\frac{1}{3} \cdot \sqrt{5} - 1\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   if 1.1e12 < x

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \frac{1}{\color{blue}{2}}\right)} \]

      4. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{\color{blue}{2}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      9. sub-to-mult N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      11. lower-unsound--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{\color{blue}{3}}{2}\right)} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      17. metadata-eval 59.6%

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      7. sub-to-mult-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \frac{1}{2}}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \frac{1}{2}\right)\right)} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right)} \]

      12. sub-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      16. add-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right)}, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      3. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      5. lower--.f64 59.5%

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot t\_2\right) \cdot t\_0\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot t\_1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot t\_1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot t\_0\right) \cdot t\_2 - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1/2 (* 1/2 (cos (* 2 x)))))
       (t_1 (- (sqrt 5) 1))
       (t_2 (* (- (cos x) 1) (sqrt 2))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+ 1 (- 1 (* (* 1/16 t_2) t_0)))
      (+ 1 (+ (* 1/2 (* (cos x) t_1)) (* 1/2 (- 3 (sqrt 5)))))))
    (if (<= x 1100000000000)
      (/
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (- (* 1/2 (* (cos y) (- (sqrt 5) 3))) (+ 1 (* 1/2 t_1))))
       3)
      (*
       1/3
       (/
        (- (* (* -1/16 t_0) t_2) -2)
        (+
         1
         (134-z0z1z2z3z4
          1/2
          t_1
          (cos x)
          (- 1 (/ 3 (sqrt 5)))
          (sqrt 5)))))))))

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(1 + 1\right) - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate--l+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}}}{3} \]

   if 1.1e12 < x

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \frac{1}{\color{blue}{2}}\right)} \]

      4. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{\color{blue}{2}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      9. sub-to-mult N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      11. lower-unsound--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{\color{blue}{3}}{2}\right)} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      17. metadata-eval 59.6%

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      7. sub-to-mult-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \frac{1}{2}}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \frac{1}{2}\right)\right)} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right)} \]

      12. sub-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      16. add-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right)}, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      3. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      5. lower--.f64 59.5%

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot t\_2\right) \cdot t\_0\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot t\_1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot t\_0\right) \cdot t\_2 - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1/2 (* 1/2 (cos (* 2 x)))))
       (t_1 (- (sqrt 5) 1))
       (t_2 (* (- (cos x) 1) (sqrt 2))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+ 1 (- 1 (* (* 1/16 t_2) t_0)))
      (+ 1 (+ (* 1/2 (* (cos x) t_1)) (* 1/2 (- 3 (sqrt 5)))))))
    (if (<= x 1100000000000)
      (*
       1/3
       (/
        (- (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- (cos y) 1)))) 2)
        (- (* 1/2 (* (cos y) (- (sqrt 5) 3))) (+ 1 (* 1/2 t_1)))))
      (*
       1/3
       (/
        (- (* (* -1/16 t_0) t_2) -2)
        (+
         1
         (134-z0z1z2z3z4
          1/2
          t_1
          (cos x)
          (- 1 (/ 3 (sqrt 5)))
          (sqrt 5)))))))))

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(1 + 1\right) - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate--l+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -1320 < x < 1.1e12

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\cos y - \cos x\right) \cdot \left(\left(\left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{1}{16} \cdot \sin y\right)\right) - 2}{\left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y - \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - -1\right)}}{3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   6. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos y - 1\right)\right)\right) - 2}{\frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right) - \left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   if 1.1e12 < x

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \frac{1}{\color{blue}{2}}\right)} \]

      4. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{\color{blue}{2}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      9. sub-to-mult N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      11. lower-unsound--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{\color{blue}{3}}{2}\right)} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      17. metadata-eval 59.6%

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      7. sub-to-mult-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \frac{1}{2}}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \frac{1}{2}\right)\right)} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right)} \]

      12. sub-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      16. add-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right)}, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      3. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      5. lower--.f64 59.5%

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -1320:\\ \;\;\;\;\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot t\_3\right) \cdot t\_1\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot t\_2\right) + \frac{1}{2} \cdot t\_0\right)}\\ \mathbf{elif}\;x \leq 1100000000000:\\ \;\;\;\;\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot t\_0\right) + \frac{1}{2} \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot t\_1\right) \cdot t\_3 - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, t\_2, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 3 (sqrt 5)))
       (t_1 (- 1/2 (* 1/2 (cos (* 2 x)))))
       (t_2 (- (sqrt 5) 1))
       (t_3 (* (- (cos x) 1) (sqrt 2))))
  (if (<= x -1320)
    (*
     1/3
     (/
      (+ 1 (- 1 (* (* 1/16 t_3) t_1)))
      (+ 1 (+ (* 1/2 (* (cos x) t_2)) (* 1/2 t_0)))))
    (if (<= x 1100000000000)
      (*
       1/3
       (/
        (+ 2 (* -1/16 (* (pow (sin y) 2) (* (sqrt 2) (- 1 (cos y))))))
        (+ 1 (+ (* 1/2 (* (cos y) t_0)) (* 1/2 t_2)))))
      (*
       1/3
       (/
        (- (* (* -1/16 t_1) t_3) -2)
        (+
         1
         (134-z0z1z2z3z4
          1/2
          t_2
          (cos x)
          (- 1 (/ 3 (sqrt 5)))
          (sqrt 5)))))))))

Derivation

1. Split input into 3 regimes

2. if x < -1320

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\left(1 + 1\right) - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. associate--l+ N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -1320 < x < 1.1e12

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\color{blue}{\sqrt{5}} - 1\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. lower-sqrt.f64 40.3%

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. Applied rewrites 40.3%

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

      3. add-flip N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

      5. lower--.f64 40.3%

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

      10. distribute-lft-out N/A

          \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

      12. lower-+.f64 40.3%

          \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

   9. Applied rewrites 40.3%

      \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   12. Applied rewrites 59.3%

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   if 1.1e12 < x

   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. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   4. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \frac{1}{\color{blue}{2}}\right)} \]

      4. mult-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{\color{blue}{2}}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

      8. div-sub N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      9. sub-to-mult N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      11. lower-unsound--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{\color{blue}{3}}{2}\right)} \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      17. metadata-eval 59.6%

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   6. Applied rewrites 59.6%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

      7. sub-to-mult-rev N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \frac{1}{2}}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \frac{1}{2}\right)\right)} \]

      11. distribute-rgt-in N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right)} \]

      12. sub-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

      16. add-flip N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right)}, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      3. add-flip N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

      5. lower--.f64 59.5%

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 59.5% accurate, 2.2× speedup

Math

\[\frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 1/3
 (/
  (-
   (*
    (* -1/16 (- 1/2 (* 1/2 (cos (* 2 x)))))
    (* (- (cos x) 1) (sqrt 2)))
   -2)
  (+
   1
   (134-z0z1z2z3z4
    1/2
    (- (sqrt 5) 1)
    (cos x)
    (- 1 (/ 3 (sqrt 5)))
    (sqrt 5))))))

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. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(3 - \sqrt{5}\right) \cdot \frac{1}{\color{blue}{2}}\right)} \]

   4. mult-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{\color{blue}{2}}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{3 - \sqrt{5}}{2}\right)} \]

   8. div-sub N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

   9. sub-to-mult N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

   10. lower-unsound-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

   11. lower-unsound--.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{\color{blue}{3}}{2}\right)} \]

   12. lower-unsound-/.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\frac{\sqrt{5}}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   13. mult-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   17. metadata-eval 59.6%

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

6. Applied rewrites 59.6%

   \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \color{blue}{\frac{3}{2}}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \color{blue}{\left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right)} \cdot \frac{3}{2}\right)} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(1 - \frac{\sqrt{5} \cdot \frac{1}{2}}{\frac{3}{2}}\right) \cdot \frac{3}{2}\right)} \]

   7. sub-to-mult-rev N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

   9. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(\frac{3}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right) \cdot \frac{1}{2}}\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{5}\right)\right)} \cdot \frac{1}{2}\right)\right)} \]

   11. distribute-rgt-in N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(3 + \left(\mathsf{neg}\left(\sqrt{5}\right)\right)\right)}\right)} \]

   12. sub-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

   13. lift--.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \frac{1}{2} \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2} + \left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

   15. distribute-rgt-out N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}} \]

   16. add-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

8. Applied rewrites 59.5%

   \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right)}, \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   3. add-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - \left(\mathsf{neg}\left(2\right)\right)}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{1 + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

   5. lower--.f64 59.5%

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]

10. Applied rewrites 59.5%

    \[\leadsto \frac{1}{3} \cdot \frac{\left(\frac{-1}{16} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) - -2}{\color{blue}{1} + \mathsf{134\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\sqrt{5} - 1\right), \cos x, \left(1 - \frac{3}{\sqrt{5}}\right), \left(\sqrt{5}\right)\right)} \]
11. Add Preprocessing

Alternative 31: 59.5% accurate, 2.3× speedup

Math

\[\frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 1/3
 (/
  (+
   1
   (-
    1
    (*
     (* 1/16 (* (- (cos x) 1) (sqrt 2)))
     (- 1/2 (* 1/2 (cos (* 2 x)))))))
  (+
   1
   (+ (* 1/2 (* (cos x) (- (sqrt 5) 1))) (* 1/2 (- 3 (sqrt 5))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + ((0.5 * (cos(x) * (sqrt(5.0) - 1.0))) + (0.5 * (3.0 - sqrt(5.0))))));
}

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 = 0.3333333333333333d0 * ((1.0d0 + (1.0d0 - ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))))) / (1.0d0 + ((0.5d0 * (cos(x) * (sqrt(5.0d0) - 1.0d0))) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))))) / (1.0 + ((0.5 * (Math.cos(x) * (Math.sqrt(5.0) - 1.0))) + (0.5 * (3.0 - Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))))) / (1.0 + ((0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))) + (0.5 * (3.0 - math.sqrt(5.0))))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(1.0 + Float64(1.0 - Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0))) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((1.0 + (1.0 - ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))))) / (1.0 + ((0.5 * (cos(x) * (sqrt(5.0) - 1.0))) + (0.5 * (3.0 - sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(1/3 * N[(N[(1 + N[(1 - N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1 + N[(N[(1/2 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 * N[(3 - N[Sqrt[5], $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. 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. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   2. add-flip N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\left(1 + 1\right) - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   4. associate--l+ N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\mathsf{neg}\left(\frac{-1}{16}\right)\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \frac{1}{16} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

6. Applied rewrites 59.5%

   \[\leadsto \frac{1}{3} \cdot \frac{1 + \left(1 - \left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)\right)}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
7. Add Preprocessing

Alternative 32: 59.5% accurate, 2.3× speedup

Math

\[\left(\left(\frac{-1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot \left(\frac{1}{\frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right) \cdot \cos x\right) - -1} \cdot \frac{1}{3}\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (-
  (*
   (* -1/16 (* (- (cos x) 1) (sqrt 2)))
   (- 1/2 (* 1/2 (cos (* 2 x)))))
  -2)
 (*
  (/ 1 (- (* 1/2 (+ (- 3 (sqrt 5)) (* (- (sqrt 5) 1) (cos x)))) -1))
  1/3)))

C

double code(double x, double y) {
	return (((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * ((1.0 / ((0.5 * ((3.0 - sqrt(5.0)) + ((sqrt(5.0) - 1.0) * cos(x)))) - -1.0)) * 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 = ((((-0.0625d0) * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - (-2.0d0)) * ((1.0d0 / ((0.5d0 * ((3.0d0 - sqrt(5.0d0)) + ((sqrt(5.0d0) - 1.0d0) * cos(x)))) - (-1.0d0))) * 0.3333333333333333d0)
end function

Java

public static double code(double x, double y) {
	return (((-0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - -2.0) * ((1.0 / ((0.5 * ((3.0 - Math.sqrt(5.0)) + ((Math.sqrt(5.0) - 1.0) * Math.cos(x)))) - -1.0)) * 0.3333333333333333);
}

Python

def code(x, y):
	return (((-0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - -2.0) * ((1.0 / ((0.5 * ((3.0 - math.sqrt(5.0)) + ((math.sqrt(5.0) - 1.0) * math.cos(x)))) - -1.0)) * 0.3333333333333333)

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - -2.0) * Float64(Float64(1.0 / Float64(Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(Float64(sqrt(5.0) - 1.0) * cos(x)))) - -1.0)) * 0.3333333333333333))
end

MATLAB

function tmp = code(x, y)
	tmp = (((-0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - -2.0) * ((1.0 / ((0.5 * ((3.0 - sqrt(5.0)) + ((sqrt(5.0) - 1.0) * cos(x)))) - -1.0)) * 0.3333333333333333);
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2), $MachinePrecision] * N[(N[(1 / N[(N[(1/2 * N[(N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision] * 1/3), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 59.5%

   \[\leadsto \left(\left(\frac{-1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - -2\right) \cdot \color{blue}{\left(\frac{1}{\frac{1}{2} \cdot \left(\left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right) \cdot \cos x\right) - -1} \cdot \frac{1}{3}\right)} \]
7. Add Preprocessing

Alternative 33: 59.5% accurate, 2.3× speedup

Math

\[\frac{\left(\left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot \frac{1}{3}}{\frac{-1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (*
  (-
   (*
    (* 1/16 (* (- (cos x) 1) (sqrt 2)))
    (- 1/2 (* 1/2 (cos (* 2 x)))))
   2)
  1/3)
 (- (* -1/2 (- (* (- (sqrt 5) 1) (cos x)) (- (sqrt 5) 3))) 1)))

C

double code(double x, double y) {
	return ((((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0) * 0.3333333333333333) / ((-0.5 * (((sqrt(5.0) - 1.0) * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
}

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 = ((((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - 2.0d0) * 0.3333333333333333d0) / (((-0.5d0) * (((sqrt(5.0d0) - 1.0d0) * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return ((((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - 2.0) * 0.3333333333333333) / ((-0.5 * (((Math.sqrt(5.0) - 1.0) * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0);
}

Python

def code(x, y):
	return ((((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - 2.0) * 0.3333333333333333) / ((-0.5 * (((math.sqrt(5.0) - 1.0) * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - 2.0) * 0.3333333333333333) / Float64(Float64(-0.5 * Float64(Float64(Float64(sqrt(5.0) - 1.0) * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0) * 0.3333333333333333) / ((-0.5 * (((sqrt(5.0) - 1.0) * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] * 1/3), $MachinePrecision] / N[(N[(-1/2 * N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 59.5%

   \[\leadsto \frac{\left(\left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot \frac{1}{3}}{\color{blue}{\frac{-1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]
7. Add Preprocessing

Alternative 34: 59.5% accurate, 2.3× speedup

Math

\[\frac{\left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - 2}{\frac{-1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \frac{1}{3} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (/
  (-
   (*
    (* 1/16 (* (- (cos x) 1) (sqrt 2)))
    (- 1/2 (* 1/2 (cos (* 2 x)))))
   2)
  (- (* -1/2 (- (* (- (sqrt 5) 1) (cos x)) (- (sqrt 5) 3))) 1))
 1/3))

C

double code(double x, double y) {
	return ((((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0) / ((-0.5 * (((sqrt(5.0) - 1.0) * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0)) * 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 = ((((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - 2.0d0) / (((-0.5d0) * (((sqrt(5.0d0) - 1.0d0) * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return ((((0.0625 * ((Math.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - 2.0) / ((-0.5 * (((Math.sqrt(5.0) - 1.0) * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0)) * 0.3333333333333333;
}

Python

def code(x, y):
	return ((((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - 2.0) / ((-0.5 * (((math.sqrt(5.0) - 1.0) * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0)) * 0.3333333333333333

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(Float64(0.0625 * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))) - 2.0) / Float64(Float64(-0.5 * Float64(Float64(Float64(sqrt(5.0) - 1.0) * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0)) * 0.3333333333333333)
end

MATLAB

function tmp = code(x, y)
	tmp = ((((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0) / ((-0.5 * (((sqrt(5.0) - 1.0) * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0)) * 0.3333333333333333;
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(1/16 * N[(N[(N[Cos[x], $MachinePrecision] - 1), $MachinePrecision] * N[Sqrt[2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1/2 - N[(1/2 * N[Cos[N[(2 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision] / N[(N[(-1/2 * N[(N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5], $MachinePrecision] - 3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] * 1/3), $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. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 59.5%

   \[\leadsto \frac{\left(\frac{1}{16} \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) - 2}{\frac{-1}{2} \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \color{blue}{\frac{1}{3}} \]
7. Add Preprocessing

Alternative 35: 42.8% accurate, 5.7× speedup

Math

\[\frac{1}{3} \cdot \frac{2}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 1/3
 (/
  2
  (+
   1
   (+ (* 1/2 (* (cos x) (- (sqrt 5) 1))) (* 1/2 (- 3 (sqrt 5))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + ((0.5 * (cos(x) * (sqrt(5.0) - 1.0))) + (0.5 * (3.0 - sqrt(5.0))))));
}

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 = 0.3333333333333333d0 * (2.0d0 / (1.0d0 + ((0.5d0 * (cos(x) * (sqrt(5.0d0) - 1.0d0))) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * (2.0 / (1.0 + ((0.5 * (Math.cos(x) * (Math.sqrt(5.0) - 1.0))) + (0.5 * (3.0 - Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return 0.3333333333333333 * (2.0 / (1.0 + ((0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))) + (0.5 * (3.0 - math.sqrt(5.0))))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(2.0 / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0))) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * (2.0 / (1.0 + ((0.5 * (cos(x) * (sqrt(5.0) - 1.0))) + (0.5 * (3.0 - sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(1/3 * N[(2 / N[(1 + N[(N[(1/2 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 * N[(3 - N[Sqrt[5], $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. 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. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 42.8%

   \[\leadsto \frac{1}{3} \cdot \frac{2}{\color{blue}{1} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 36: 40.3% accurate, 19.2× speedup

Math

\[\frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/ 2/3 (- (* 1/2 (+ (- (sqrt 5) 1) (- 3 (sqrt 5)))) -1)))

C

double code(double x, double y) {
	return 0.6666666666666666 / ((0.5 * ((sqrt(5.0) - 1.0) + (3.0 - sqrt(5.0)))) - -1.0);
}

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 = 0.6666666666666666d0 / ((0.5d0 * ((sqrt(5.0d0) - 1.0d0) + (3.0d0 - sqrt(5.0d0)))) - (-1.0d0))
end function

Java

public static double code(double x, double y) {
	return 0.6666666666666666 / ((0.5 * ((Math.sqrt(5.0) - 1.0) + (3.0 - Math.sqrt(5.0)))) - -1.0);
}

Python

def code(x, y):
	return 0.6666666666666666 / ((0.5 * ((math.sqrt(5.0) - 1.0) + (3.0 - math.sqrt(5.0)))) - -1.0)

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(Float64(0.5 * Float64(Float64(sqrt(5.0) - 1.0) + Float64(3.0 - sqrt(5.0)))) - -1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.6666666666666666 / ((0.5 * ((sqrt(5.0) - 1.0) + (3.0 - sqrt(5.0)))) - -1.0);
end

Wolfram

code[x_, y_] := N[(2/3 / N[(N[(1/2 * N[(N[(N[Sqrt[5], $MachinePrecision] - 1), $MachinePrecision] + N[(3 - N[Sqrt[5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

4. Applied rewrites 59.6%

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\color{blue}{\sqrt{5}} - 1\right)\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

   9. lower-sqrt.f64 40.3%

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 40.3%

   \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1} \]

   3. add-flip N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

   5. lower--.f64 40.3%

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) - -1} \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) - -1} \]

   10. distribute-lft-out N/A

       \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

   12. lower-+.f64 40.3%

       \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]

9. Applied rewrites 40.3%

   \[\leadsto \frac{\frac{2}{3}}{\frac{1}{2} \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) - -1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))