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

Percentage Accurate: 99.3% → 99.3%

Time: 9.6s

Alternatives: 37

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) - cos(y)))) / (3.0d0 + ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 3.0d0))
end function

Java

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

Python

def code(x, y):
	return (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) - math.cos(y)))) / (3.0 + ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 3.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

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

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

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

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

   5. lower-unsound-/.f64 99.3%

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

5. Applied rewrites 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(0.5 \cdot \left(2.23606797749979 - 1\right)\right) \cdot \cos x - \left(\left(2.23606797749979 - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/
 (+
  2.0
  (*
   (- (sin y) (* 0.0625 (sin x)))
   (*
    (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
    (- (cos x) (cos y)))))
 (+
  3.0
  (*
   (-
    (* (* 0.5 (- 2.23606797749979 1.0)) (cos x))
    (* (* (- 2.23606797749979 3.0) 0.5) (cos y)))
   3.0))))

C

double code(double x, double y) {
	return (2.0 + ((sin(y) - (0.0625 * sin(x))) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y))))) / (3.0 + ((((0.5 * (2.23606797749979 - 1.0)) * cos(x)) - (((2.23606797749979 - 3.0) * 0.5) * cos(y))) * 3.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * (cos(x) - cos(y))))) / (3.0d0 + ((((0.5d0 * (2.23606797749979d0 - 1.0d0)) * cos(x)) - (((2.23606797749979d0 - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * (Math.cos(x) - Math.cos(y))))) / (3.0 + ((((0.5 * (2.23606797749979 - 1.0)) * Math.cos(x)) - (((2.23606797749979 - 3.0) * 0.5) * Math.cos(y))) * 3.0));
}

Python

def code(x, y):
	return (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * (math.cos(x) - math.cos(y))))) / (3.0 + ((((0.5 * (2.23606797749979 - 1.0)) * math.cos(x)) - (((2.23606797749979 - 3.0) * 0.5) * math.cos(y))) * 3.0))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * Float64(2.23606797749979 - 1.0)) * cos(x)) - Float64(Float64(Float64(2.23606797749979 - 3.0) * 0.5) * cos(y))) * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y))))) / (3.0 + ((((0.5 * (2.23606797749979 - 1.0)) * cos(x)) - (((2.23606797749979 - 3.0) * 0.5) * cos(y))) * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[(0.5 * N[(2.23606797749979 - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(2.23606797749979 - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. mult-flip N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 99.3%

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

5. Applied rewrites 99.3%

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

6. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(0.5 \cdot \left(\color{blue}{2.23606797749979} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

7. Evaluated real constant 99.3%

   \[\leadsto \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(0.5 \cdot \left(2.23606797749979 - 1\right)\right) \cdot \cos x - \left(\left(\color{blue}{2.23606797749979} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
8. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(2.23606797749979 - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - 2.23606797749979\right)\right)\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 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) (- 2.23606797749979 1.0)))
    (* 0.5 (* (cos y) (- 3.0 2.23606797749979))))))))

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) * (2.23606797749979 - 1.0))) + (0.5 * (cos(y) * (3.0 - 2.23606797749979))))));
}

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) * (2.23606797749979d0 - 1.0d0))) + (0.5d0 * (cos(y) * (3.0d0 - 2.23606797749979d0))))))
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) * (2.23606797749979 - 1.0))) + (0.5 * (Math.cos(y) * (3.0 - 2.23606797749979))))));
}

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) * (2.23606797749979 - 1.0))) + (0.5 * (math.cos(y) * (3.0 - 2.23606797749979))))))

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(2.23606797749979 - 1.0))) + Float64(0.5 * Float64(cos(y) * Float64(3.0 - 2.23606797749979)))))))
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) * (2.23606797749979 - 1.0))) + (0.5 * (cos(y) * (3.0 - 2.23606797749979))))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(2.23606797749979 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - 2.23606797749979), $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 x around inf

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

4. Applied rewrites 99.2%

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

5. Evaluated real constant 99.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(2.23606797749979 - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

6. Evaluated real constant 99.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(2.23606797749979 - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - 2.23606797749979\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x\\ t_2 := \frac{2 + \sin y \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0\right)}{3 + \left(t\_1 - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;y \leq -3.8:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.9:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - 0.0625 \cdot y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{3 + \left(t\_1 - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1 (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x)))
       (t_2
        (/
         (+
          2.0
          (*
           (sin y)
           (* (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0)) t_0)))
         (+
          3.0
          (* (- t_1 (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y))) 3.0)))))
  (if (<= y -3.8)
    t_2
    (if (<= y 2.9)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (* 0.0625 y)))
          (- (sin y) (/ (sin x) 16.0)))
         t_0))
       (+
        3.0
        (*
         (-
          t_1
          (*
           (* (* (- 1.0 (/ 3.0 (sqrt 5.0))) (sqrt 5.0)) 0.5)
           (cos y)))
         3.0)))
      t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (0.5 * (sqrt(5.0) - 1.0)) * cos(x);
	double t_2 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / (3.0 + ((t_1 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (y <= -3.8) {
		tmp = t_2;
	} else if (y <= 2.9) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / (3.0 + ((t_1 - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	} 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 = cos(x) - cos(y)
    t_1 = (0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)
    t_2 = (2.0d0 + (sin(y) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * t_0))) / (3.0d0 + ((t_1 - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    if (y <= (-3.8d0)) then
        tmp = t_2
    else if (y <= 2.9d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (0.0625d0 * y))) * (sin(y) - (sin(x) / 16.0d0))) * t_0)) / (3.0d0 + ((t_1 - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 3.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = (0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x);
	double t_2 = (2.0 + (Math.sin(y) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * t_0))) / (3.0 + ((t_1 - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (y <= -3.8) {
		tmp = t_2;
	} else if (y <= 2.9) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (0.0625 * y))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * t_0)) / (3.0 + ((t_1 - ((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y))) * 3.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = (0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)
	t_2 = (2.0 + (math.sin(y) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * t_0))) / (3.0 + ((t_1 - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if y <= -3.8:
		tmp = t_2
	elif y <= 2.9:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (0.0625 * y))) * (math.sin(y) - (math.sin(x) / 16.0))) * t_0)) / (3.0 + ((t_1 - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 3.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x))
	t_2 = Float64(Float64(2.0 + Float64(sin(y) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / Float64(3.0 + Float64(Float64(t_1 - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (y <= -3.8)
		tmp = t_2;
	elseif (y <= 2.9)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(0.0625 * y))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / Float64(3.0 + Float64(Float64(t_1 - Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = (0.5 * (sqrt(5.0) - 1.0)) * cos(x);
	t_2 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / (3.0 + ((t_1 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (y <= -3.8)
		tmp = t_2;
	elseif (y <= 2.9)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (0.0625 * y))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / (3.0 + ((t_1 - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[Sin[y], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(t$95$1 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8], t$95$2, If[LessEqual[y, 2.9], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(t$95$1 - N[(N[(N[(N[(1.0 - N[(3.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998 or 2.8999999999999999 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   8. Applied rewrites 64.3%

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

   if -3.7999999999999998 < y < 2.8999999999999999

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

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

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

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

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 50.3%

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

   8. Applied rewrites 50.3%

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

Alternative 5: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3\\ t_2 := \frac{2 + \sin y \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0\right)}{t\_1}\\ \mathbf{if}\;y \leq -3.8:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.9:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot y\right) \cdot \sqrt{2}\right) \cdot t\_0\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1
        (+
         3.0
         (*
          (-
           (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x))
           (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
          3.0)))
       (t_2
        (/
         (+
          2.0
          (*
           (sin y)
           (* (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0)) t_0)))
         t_1)))
  (if (<= y -3.8)
    t_2
    (if (<= y 2.9)
      (/
       (+
        2.0
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (* (* (- (sin x) (* 0.0625 y)) (sqrt 2.0)) t_0)))
       t_1)
      t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	double t_2 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1;
	double tmp;
	if (y <= -3.8) {
		tmp = t_2;
	} else if (y <= 2.9) {
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (((sin(x) - (0.0625 * y)) * sqrt(2.0)) * t_0))) / t_1;
	} 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 = cos(x) - cos(y)
    t_1 = 3.0d0 + ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0)
    t_2 = (2.0d0 + (sin(y) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * t_0))) / t_1
    if (y <= (-3.8d0)) then
        tmp = t_2
    else if (y <= 2.9d0) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (((sin(x) - (0.0625d0 * y)) * sqrt(2.0d0)) * t_0))) / t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = 3.0 + ((((0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0);
	double t_2 = (2.0 + (Math.sin(y) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * t_0))) / t_1;
	double tmp;
	if (y <= -3.8) {
		tmp = t_2;
	} else if (y <= 2.9) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (((Math.sin(x) - (0.0625 * y)) * Math.sqrt(2.0)) * t_0))) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = 3.0 + ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0)
	t_2 = (2.0 + (math.sin(y) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * t_0))) / t_1
	tmp = 0
	if y <= -3.8:
		tmp = t_2
	elif y <= 2.9:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (((math.sin(x) - (0.0625 * y)) * math.sqrt(2.0)) * t_0))) / t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0))
	t_2 = Float64(Float64(2.0 + Float64(sin(y) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1)
	tmp = 0.0
	if (y <= -3.8)
		tmp = t_2;
	elseif (y <= 2.9)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * y)) * sqrt(2.0)) * t_0))) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	t_2 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1;
	tmp = 0.0;
	if (y <= -3.8)
		tmp = t_2;
	elseif (y <= 2.9)
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (((sin(x) - (0.0625 * y)) * sqrt(2.0)) * t_0))) / t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(N[(N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[Sin[y], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[y, -3.8], t$95$2, If[LessEqual[y, 2.9], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998 or 2.8999999999999999 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   8. Applied rewrites 64.3%

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

   if -3.7999999999999998 < y < 2.8999999999999999

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - \color{blue}{\frac{1}{16} \cdot y}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 50.2%

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

   8. Applied rewrites 50.2%

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

Alternative 6: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3\\ t_2 := \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot t\_0\right)}{t\_1}\\ \mathbf{if}\;x \leq -0.035:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot t\_0\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1
        (+
         3.0
         (*
          (-
           (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x))
           (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
          3.0)))
       (t_2
        (/
         (+
          2.0
          (*
           (- (sin y) (* 0.0625 (sin x)))
           (* (* (sin x) (sqrt 2.0)) t_0)))
         t_1)))
  (if (<= x -0.035)
    t_2
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (- (sin y) (* 0.0625 x))
         (* (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0)) t_0)))
       t_1)
      t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	double t_2 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * ((sin(x) * sqrt(2.0)) * t_0))) / t_1;
	double tmp;
	if (x <= -0.035) {
		tmp = t_2;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + ((sin(y) - (0.0625 * x)) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1;
	} 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 = cos(x) - cos(y)
    t_1 = 3.0d0 + ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0)
    t_2 = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * ((sin(x) * sqrt(2.0d0)) * t_0))) / t_1
    if (x <= (-0.035d0)) then
        tmp = t_2
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * x)) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * t_0))) / t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = 3.0 + ((((0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0);
	double t_2 = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * ((Math.sin(x) * Math.sqrt(2.0)) * t_0))) / t_1;
	double tmp;
	if (x <= -0.035) {
		tmp = t_2;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * x)) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * t_0))) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = 3.0 + ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0)
	t_2 = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * ((math.sin(x) * math.sqrt(2.0)) * t_0))) / t_1
	tmp = 0
	if x <= -0.035:
		tmp = t_2
	elif x <= 1.45e-5:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * x)) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * t_0))) / t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(Float64(sin(x) * sqrt(2.0)) * t_0))) / t_1)
	tmp = 0.0
	if (x <= -0.035)
		tmp = t_2;
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * x)) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	t_2 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * ((sin(x) * sqrt(2.0)) * t_0))) / t_1;
	tmp = 0.0;
	if (x <= -0.035)
		tmp = t_2;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + ((sin(y) - (0.0625 * x)) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * t_0))) / t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[(N[(N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[x, -0.035], t$95$2, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.035000000000000003 or 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.4%

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

   8. Applied rewrites 63.4%

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

   if -0.035000000000000003 < x < 1.45e-5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 51.5%

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

   8. Applied rewrites 51.5%

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

Alternative 7: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := \left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x\\ t_2 := \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\\ t_3 := \frac{2 + \sin y \cdot t\_2}{3 + \left(t\_1 - \left(t\_0 \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot t\_2}{3 + \left(t\_1 - 0.5 \cdot t\_0\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 3.0))
       (t_1 (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x)))
       (t_2
        (*
         (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
         (- (cos x) (cos y))))
       (t_3
        (/
         (+ 2.0 (* (sin y) t_2))
         (+ 3.0 (* (- t_1 (* (* t_0 0.5) (cos y))) 3.0)))))
  (if (<= y -19.5)
    t_3
    (if (<= y 2.16e-7)
      (/
       (+ 2.0 (* (- (sin y) (* 0.0625 (sin x))) t_2))
       (+ 3.0 (* (- t_1 (* 0.5 t_0)) 3.0)))
      t_3))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = (0.5 * (sqrt(5.0) - 1.0)) * cos(x);
	double t_2 = ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y));
	double t_3 = (2.0 + (sin(y) * t_2)) / (3.0 + ((t_1 - ((t_0 * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_3;
	} else if (y <= 2.16e-7) {
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * t_2)) / (3.0 + ((t_1 - (0.5 * t_0)) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) - 3.0d0
    t_1 = (0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)
    t_2 = ((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * (cos(x) - cos(y))
    t_3 = (2.0d0 + (sin(y) * t_2)) / (3.0d0 + ((t_1 - ((t_0 * 0.5d0) * cos(y))) * 3.0d0))
    if (y <= (-19.5d0)) then
        tmp = t_3
    else if (y <= 2.16d-7) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * t_2)) / (3.0d0 + ((t_1 - (0.5d0 * t_0)) * 3.0d0))
    else
        tmp = t_3
    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 = (0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x);
	double t_2 = ((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * (Math.cos(x) - Math.cos(y));
	double t_3 = (2.0 + (Math.sin(y) * t_2)) / (3.0 + ((t_1 - ((t_0 * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_3;
	} else if (y <= 2.16e-7) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * t_2)) / (3.0 + ((t_1 - (0.5 * t_0)) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 3.0
	t_1 = (0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)
	t_2 = ((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * (math.cos(x) - math.cos(y))
	t_3 = (2.0 + (math.sin(y) * t_2)) / (3.0 + ((t_1 - ((t_0 * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if y <= -19.5:
		tmp = t_3
	elif y <= 2.16e-7:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * t_2)) / (3.0 + ((t_1 - (0.5 * t_0)) * 3.0))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x))
	t_2 = Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(cos(x) - cos(y)))
	t_3 = Float64(Float64(2.0 + Float64(sin(y) * t_2)) / Float64(3.0 + Float64(Float64(t_1 - Float64(Float64(t_0 * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (y <= -19.5)
		tmp = t_3;
	elseif (y <= 2.16e-7)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * t_2)) / Float64(3.0 + Float64(Float64(t_1 - Float64(0.5 * t_0)) * 3.0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 3.0;
	t_1 = (0.5 * (sqrt(5.0) - 1.0)) * cos(x);
	t_2 = ((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y));
	t_3 = (2.0 + (sin(y) * t_2)) / (3.0 + ((t_1 - ((t_0 * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (y <= -19.5)
		tmp = t_3;
	elseif (y <= 2.16e-7)
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * t_2)) / (3.0 + ((t_1 - (0.5 * t_0)) * 3.0));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(t$95$1 - N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -19.5], t$95$3, If[LessEqual[y, 2.16e-7], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(t$95$1 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -19.5 or 2.16e-7 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   8. Applied rewrites 64.3%

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

   if -19.5 < y < 2.16e-7

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-sqrt.f64 59.7%

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

   8. Applied rewrites 59.7%

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

Alternative 8: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin x - 0.0625 \cdot \sin y\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{2 + \sin y \cdot \left(\left(t\_1 \cdot \sqrt{2}\right) \cdot t\_0\right)}{3 + \left(\left(0.5 \cdot t\_2\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(t\_0 \cdot \left(t\_1 \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1 (- (sin x) (* 0.0625 (sin y))))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3
        (/
         (+ 2.0 (* (sin y) (* (* t_1 (sqrt 2.0)) t_0)))
         (+
          3.0
          (*
           (-
            (* (* 0.5 t_2) (cos x))
            (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
           3.0)))))
  (if (<= y -19.5)
    t_3
    (if (<= y 2.16e-7)
      (*
       0.3333333333333333
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (* t_0 (* t_1 (- (sin y) (* 0.0625 (sin x)))))))
        (+
         1.0
         (+ (* 0.5 (* (cos x) t_2)) (* 0.5 (- 3.0 (sqrt 5.0)))))))
      t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sin(x) - (0.0625 * sin(y));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = (2.0 + (sin(y) * ((t_1 * sqrt(2.0)) * t_0))) / (3.0 + ((((0.5 * t_2) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_3;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_0 * (t_1 * (sin(y) - (0.0625 * sin(x))))))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * (3.0 - sqrt(5.0))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = sin(x) - (0.0625d0 * sin(y))
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = (2.0d0 + (sin(y) * ((t_1 * sqrt(2.0d0)) * t_0))) / (3.0d0 + ((((0.5d0 * t_2) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    if (y <= (-19.5d0)) then
        tmp = t_3
    else if (y <= 2.16d-7) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * (t_0 * (t_1 * (sin(y) - (0.0625d0 * sin(x))))))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sin(x) - (0.0625 * Math.sin(y));
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = (2.0 + (Math.sin(y) * ((t_1 * Math.sqrt(2.0)) * t_0))) / (3.0 + ((((0.5 * t_2) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_3;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * (t_0 * (t_1 * (Math.sin(y) - (0.0625 * Math.sin(x))))))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + (0.5 * (3.0 - Math.sqrt(5.0))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sin(x) - (0.0625 * math.sin(y))
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = (2.0 + (math.sin(y) * ((t_1 * math.sqrt(2.0)) * t_0))) / (3.0 + ((((0.5 * t_2) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if y <= -19.5:
		tmp = t_3
	elif y <= 2.16e-7:
		tmp = 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * (t_0 * (t_1 * (math.sin(y) - (0.0625 * math.sin(x))))))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + (0.5 * (3.0 - math.sqrt(5.0))))))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sin(x) - Float64(0.0625 * sin(y)))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(2.0 + Float64(sin(y) * Float64(Float64(t_1 * sqrt(2.0)) * t_0))) / Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * t_2) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (y <= -19.5)
		tmp = t_3;
	elseif (y <= 2.16e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_0 * Float64(t_1 * Float64(sin(y) - Float64(0.0625 * sin(x))))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sin(x) - (0.0625 * sin(y));
	t_2 = sqrt(5.0) - 1.0;
	t_3 = (2.0 + (sin(y) * ((t_1 * sqrt(2.0)) * t_0))) / (3.0 + ((((0.5 * t_2) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (y <= -19.5)
		tmp = t_3;
	elseif (y <= 2.16e-7)
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (t_0 * (t_1 * (sin(y) - (0.0625 * sin(x))))))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * (3.0 - sqrt(5.0))))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[Sin[y], $MachinePrecision] * N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -19.5], t$95$3, If[LessEqual[y, 2.16e-7], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -19.5 or 2.16e-7 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   8. Applied rewrites 64.3%

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

   if -19.5 < y < 2.16e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 59.7%

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

   7. Applied rewrites 59.7%

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

Alternative 9: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(t\_0 \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 + \left(t\_0 \cdot 1 - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_1
        (/
         (+
          2.0
          (*
           (- (sin y) (* 0.0625 (sin x)))
           (* (* (sin x) (sqrt 2.0)) (- (cos x) (cos y)))))
         (+
          3.0
          (*
           (- (* t_0 (cos x)) (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
           3.0)))))
  (if (<= x -175000000.0)
    t_1
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (/ (sin x) 16.0)))
         (- 1.0 (cos y))))
       (+
        3.0
        (*
         (-
          (* t_0 1.0)
          (*
           (* (* (- 1.0 (/ 3.0 (sqrt 5.0))) (sqrt 5.0)) 0.5)
           (cos y)))
         3.0)))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * ((sin(x) * sqrt(2.0)) * (cos(x) - cos(y))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (x <= -175000000.0) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * ((sin(x) * sqrt(2.0d0)) * (cos(x) - cos(y))))) / (3.0d0 + (((t_0 * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    if (x <= (-175000000.0d0)) then
        tmp = t_1
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (1.0d0 - cos(y)))) / (3.0d0 + (((t_0 * 1.0d0) - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 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 = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * ((Math.sin(x) * Math.sqrt(2.0)) * (Math.cos(x) - Math.cos(y))))) / (3.0 + (((t_0 * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (x <= -175000000.0) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (1.0 - Math.cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y))) * 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 = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * ((math.sin(x) * math.sqrt(2.0)) * (math.cos(x) - math.cos(y))))) / (3.0 + (((t_0 * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if x <= -175000000.0:
		tmp = t_1
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (1.0 - math.cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 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(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(cos(x) - cos(y))))) / Float64(3.0 + Float64(Float64(Float64(t_0 * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = 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(1.0 - cos(y)))) / Float64(3.0 + Float64(Float64(Float64(t_0 * 1.0) - Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * ((sin(x) * sqrt(2.0)) * (cos(x) - cos(y))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], t$95$1, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[(N[(N[(N[(1.0 - N[(3.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e8 or 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.4%

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

   8. Applied rewrites 63.4%

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

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

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

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

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

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.1%

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

Alternative 10: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3\\ t_1 := \frac{2 + \sin y \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.046:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.034:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (+
         3.0
         (*
          (-
           (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x))
           (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
          3.0)))
       (t_1
        (/
         (+
          2.0
          (*
           (sin y)
           (*
            (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
            (- (cos x) (cos y)))))
         t_0)))
  (if (<= y -0.046)
    t_1
    (if (<= y 0.034)
      (/
       (+
        2.0
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
       t_0)
      t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	double t_1 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y))))) / t_0;
	double tmp;
	if (y <= -0.046) {
		tmp = t_1;
	} else if (y <= 0.034) {
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_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 = 3.0d0 + ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0)
    t_1 = (2.0d0 + (sin(y) * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * (cos(x) - cos(y))))) / t_0
    if (y <= (-0.046d0)) then
        tmp = t_1
    else if (y <= 0.034d0) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + ((((0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0);
	double t_1 = (2.0 + (Math.sin(y) * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * (Math.cos(x) - Math.cos(y))))) / t_0;
	double tmp;
	if (y <= -0.046) {
		tmp = t_1;
	} else if (y <= 0.034) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0)
	t_1 = (2.0 + (math.sin(y) * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * (math.cos(x) - math.cos(y))))) / t_0
	tmp = 0
	if y <= -0.046:
		tmp = t_1
	elif y <= 0.034:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0))
	t_1 = Float64(Float64(2.0 + Float64(sin(y) * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(cos(x) - cos(y))))) / t_0)
	tmp = 0.0
	if (y <= -0.046)
		tmp = t_1;
	elseif (y <= 0.034)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0);
	t_1 = (2.0 + (sin(y) * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (cos(x) - cos(y))))) / t_0;
	tmp = 0.0;
	if (y <= -0.046)
		tmp = t_1;
	elseif (y <= 0.034)
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(N[(N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[Sin[y], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.046], t$95$1, If[LessEqual[y, 0.034], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.045999999999999999 or 0.034000000000000002 < y

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   8. Applied rewrites 64.3%

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

   if -0.045999999999999999 < y < 0.034000000000000002

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

Alternative 11: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -0.046:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot t\_0}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - 2.23606797749979\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - 2.23606797749979\right) \cdot 0.5\right) \cdot \cos x\right)}\\ \mathbf{elif}\;y \leq 0.034:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(0.5 \cdot t\_1\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\sin y \cdot \sqrt{2}\right) \cdot \left(\sin x - 0.0625 \cdot \sin y\right)\right) \cdot t\_0 - -2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{1 + 0.5 \cdot \left(\cos x \cdot t\_1 + \left(3 - \sqrt{5}\right) \cdot \cos y\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y))) (t_1 (- (sqrt 5.0) 1.0)))
  (if (<= y -0.046)
    (/
     (+
      2.0
      (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y)) t_0))
     (*
      3.0
      (-
       (+ (* (* 0.5 (- 3.0 2.23606797749979)) (cos y)) 1.0)
       (* (* (- 1.0 2.23606797749979) 0.5) (cos x)))))
    (if (<= y 0.034)
      (/
       (+
        2.0
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
       (+
        3.0
        (*
         (-
          (* (* 0.5 t_1) (cos x))
          (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
         3.0)))
      (*
       (*
        (-
         (*
          (* (* (sin y) (sqrt 2.0)) (- (sin x) (* 0.0625 (sin y))))
          t_0)
         -2.0)
        0.3333333333333333)
       (/
        1.0
        (+
         1.0
         (*
          0.5
          (+ (* (cos x) t_1) (* (- 3.0 (sqrt 5.0)) (cos y)))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.046) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * cos(x))));
	} else if (y <= 0.034) {
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + ((((0.5 * t_1) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	} else {
		tmp = (((((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y)))) * t_0) - -2.0) * 0.3333333333333333) * (1.0 / (1.0 + (0.5 * ((cos(x) * t_1) + ((3.0 - sqrt(5.0)) * 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) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = sqrt(5.0d0) - 1.0d0
    if (y <= (-0.046d0)) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * sin(y)) * t_0)) / (3.0d0 * ((((0.5d0 * (3.0d0 - 2.23606797749979d0)) * cos(y)) + 1.0d0) - (((1.0d0 - 2.23606797749979d0) * 0.5d0) * cos(x))))
    else if (y <= 0.034d0) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + ((((0.5d0 * t_1) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    else
        tmp = (((((sin(y) * sqrt(2.0d0)) * (sin(x) - (0.0625d0 * sin(y)))) * t_0) - (-2.0d0)) * 0.3333333333333333d0) * (1.0d0 / (1.0d0 + (0.5d0 * ((cos(x) * t_1) + ((3.0d0 - sqrt(5.0d0)) * cos(y))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -0.046) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * Math.sin(y)) * t_0)) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * Math.cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * Math.cos(x))));
	} else if (y <= 0.034) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + ((((0.5 * t_1) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	} else {
		tmp = (((((Math.sin(y) * Math.sqrt(2.0)) * (Math.sin(x) - (0.0625 * Math.sin(y)))) * t_0) - -2.0) * 0.3333333333333333) * (1.0 / (1.0 + (0.5 * ((Math.cos(x) * t_1) + ((3.0 - Math.sqrt(5.0)) * Math.cos(y))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sqrt(5.0) - 1.0
	tmp = 0
	if y <= -0.046:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * math.sin(y)) * t_0)) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * math.cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * math.cos(x))))
	elif y <= 0.034:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + ((((0.5 * t_1) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	else:
		tmp = (((((math.sin(y) * math.sqrt(2.0)) * (math.sin(x) - (0.0625 * math.sin(y)))) * t_0) - -2.0) * 0.3333333333333333) * (1.0 / (1.0 + (0.5 * ((math.cos(x) * t_1) + ((3.0 - math.sqrt(5.0)) * math.cos(y))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -0.046)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * t_0)) / Float64(3.0 * Float64(Float64(Float64(Float64(0.5 * Float64(3.0 - 2.23606797749979)) * cos(y)) + 1.0) - Float64(Float64(Float64(1.0 - 2.23606797749979) * 0.5) * cos(x)))));
	elseif (y <= 0.034)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * t_1) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sin(y) * sqrt(2.0)) * Float64(sin(x) - Float64(0.0625 * sin(y)))) * t_0) - -2.0) * 0.3333333333333333) * Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_1) + Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sqrt(5.0) - 1.0;
	tmp = 0.0;
	if (y <= -0.046)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * t_0)) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * cos(x))));
	elseif (y <= 0.034)
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + ((((0.5 * t_1) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	else
		tmp = (((((sin(y) * sqrt(2.0)) * (sin(x) - (0.0625 * sin(y)))) * t_0) - -2.0) * 0.3333333333333333) * (1.0 / (1.0 + (0.5 * ((cos(x) * t_1) + ((3.0 - sqrt(5.0)) * cos(y))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -0.046], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(0.5 * N[(3.0 - 2.23606797749979), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(1.0 - 2.23606797749979), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.034], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.045999999999999999

   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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   6. Applied rewrites 64.3%

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

   7. Evaluated real constant 64.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - \color{blue}{2.23606797749979}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)} \]

   8. Evaluated real constant 64.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - 2.23606797749979\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \color{blue}{2.23606797749979}\right) \cdot 0.5\right) \cdot \cos x\right)} \]

   if -0.045999999999999999 < y < 0.034000000000000002

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

   if 0.034000000000000002 < y

   1. Initial program 99.3%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   6. Applied rewrites 64.3%

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

   8. Applied rewrites 64.2%

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

Alternative 12: 80.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - 2.23606797749979\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - 2.23606797749979\right) \cdot 0.5\right) \cdot \cos x\right)}\\ \mathbf{if}\;y \leq -0.046:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.034:\\ \;\;\;\;\frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0
        (/
         (+
          2.0
          (*
           (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))
           (- (cos x) (cos y))))
         (*
          3.0
          (-
           (+ (* (* 0.5 (- 3.0 2.23606797749979)) (cos y)) 1.0)
           (* (* (- 1.0 2.23606797749979) 0.5) (cos x)))))))
  (if (<= y -0.046)
    t_0
    (if (<= y 0.034)
      (/
       (+
        2.0
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
       (+
        3.0
        (*
         (-
          (* (* 0.5 (- (sqrt 5.0) 1.0)) (cos x))
          (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
         3.0)))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * cos(x))));
	double tmp;
	if (y <= -0.046) {
		tmp = t_0;
	} else if (y <= 0.034) {
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	} else {
		tmp = 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) :: tmp
    t_0 = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0d0 * ((((0.5d0 * (3.0d0 - 2.23606797749979d0)) * cos(y)) + 1.0d0) - (((1.0d0 - 2.23606797749979d0) * 0.5d0) * cos(x))))
    if (y <= (-0.046d0)) then
        tmp = t_0
    else if (y <= 0.034d0) then
        tmp = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + ((((0.5d0 * (sqrt(5.0d0) - 1.0d0)) * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * Math.sin(y)) * (Math.cos(x) - Math.cos(y)))) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * Math.cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * Math.cos(x))));
	double tmp;
	if (y <= -0.046) {
		tmp = t_0;
	} else if (y <= 0.034) {
		tmp = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + ((((0.5 * (Math.sqrt(5.0) - 1.0)) * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * math.sin(y)) * (math.cos(x) - math.cos(y)))) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * math.cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * math.cos(x))))
	tmp = 0
	if y <= -0.046:
		tmp = t_0
	elif y <= 0.034:
		tmp = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + ((((0.5 * (math.sqrt(5.0) - 1.0)) * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(Float64(Float64(0.5 * Float64(3.0 - 2.23606797749979)) * cos(y)) + 1.0) - Float64(Float64(Float64(1.0 - 2.23606797749979) * 0.5) * cos(x)))))
	tmp = 0.0
	if (y <= -0.046)
		tmp = t_0;
	elseif (y <= 0.034)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(Float64(0.5 * Float64(sqrt(5.0) - 1.0)) * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * ((((0.5 * (3.0 - 2.23606797749979)) * cos(y)) + 1.0) - (((1.0 - 2.23606797749979) * 0.5) * cos(x))));
	tmp = 0.0;
	if (y <= -0.046)
		tmp = t_0;
	elseif (y <= 0.034)
		tmp = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + ((((0.5 * (sqrt(5.0) - 1.0)) * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(0.5 * N[(3.0 - 2.23606797749979), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(1.0 - 2.23606797749979), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.046], t$95$0, If[LessEqual[y, 0.034], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.045999999999999999 or 0.034000000000000002 < y

   1. Initial program 99.3%

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

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-sin.f64 64.3%

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

   6. Applied rewrites 64.3%

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

   7. Evaluated real constant 64.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - \color{blue}{2.23606797749979}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)} \]

   8. Evaluated real constant 64.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - 2.23606797749979\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \color{blue}{2.23606797749979}\right) \cdot 0.5\right) \cdot \cos x\right)} \]

   if -0.045999999999999999 < y < 0.034000000000000002

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

Alternative 13: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(t\_0 \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 + \left(t\_0 \cdot 1 - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_1
        (/
         (+
          2.0
          (*
           (- (sin y) (* 0.0625 (sin x)))
           (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+
          3.0
          (*
           (- (* t_0 (cos x)) (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
           3.0)))))
  (if (<= x -2.3e+27)
    t_1
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (- (sin y) (/ (sin x) 16.0)))
         (- 1.0 (cos y))))
       (+
        3.0
        (*
         (-
          (* t_0 1.0)
          (*
           (* (* (- 1.0 (/ 3.0 (sqrt 5.0))) (sqrt 5.0)) 0.5)
           (cos y)))
         3.0)))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (x <= -2.3e+27) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (((t_0 * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    if (x <= (-2.3d+27)) then
        tmp = t_1
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (1.0d0 - cos(y)))) / (3.0d0 + (((t_0 * 1.0d0) - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 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 = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (((t_0 * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (x <= -2.3e+27) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (1.0 - Math.cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y))) * 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 = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (((t_0 * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if x <= -2.3e+27:
		tmp = t_1
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (1.0 - math.cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 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(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(t_0 * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (x <= -2.3e+27)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = 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(1.0 - cos(y)))) / Float64(3.0 + Float64(Float64(Float64(t_0 * 1.0) - Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (x <= -2.3e+27)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / (3.0 + (((t_0 * 1.0) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e+27], t$95$1, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[(N[(N[(N[(1.0 - N[(3.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e27 or 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

   if -2.3000000000000001e27 < x < 1.45e-5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

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

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

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

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.1%

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

Alternative 14: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\\ t_1 := \sin y - 0.0625 \cdot \sin x\\ t_2 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_3 := \frac{2 + t\_1 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(t\_2 \cdot \cos x - t\_0\right) \cdot 3}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+27}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)\right)}{3 + \left(t\_2 \cdot 1 - t\_0\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
       (t_1 (- (sin y) (* 0.0625 (sin x))))
       (t_2 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_3
        (/
         (+ 2.0 (* t_1 (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+ 3.0 (* (- (* t_2 (cos x)) t_0) 3.0)))))
  (if (<= x -2.3e+27)
    t_3
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         t_1
         (*
          (* (- (sin x) (* 0.0625 (sin y))) (sqrt 2.0))
          (- 1.0 (cos y)))))
       (+ 3.0 (* (- (* t_2 1.0) t_0) 3.0)))
      t_3))))

C

double code(double x, double y) {
	double t_0 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	double t_1 = sin(y) - (0.0625 * sin(x));
	double t_2 = 0.5 * (sqrt(5.0) - 1.0);
	double t_3 = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_2 * cos(x)) - t_0) * 3.0));
	double tmp;
	if (x <= -2.3e+27) {
		tmp = t_3;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (t_1 * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (1.0 - cos(y))))) / (3.0 + (((t_2 * 1.0) - t_0) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = ((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y)
    t_1 = sin(y) - (0.0625d0 * sin(x))
    t_2 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_3 = (2.0d0 + (t_1 * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (((t_2 * cos(x)) - t_0) * 3.0d0))
    if (x <= (-2.3d+27)) then
        tmp = t_3
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (t_1 * (((sin(x) - (0.0625d0 * sin(y))) * sqrt(2.0d0)) * (1.0d0 - cos(y))))) / (3.0d0 + (((t_2 * 1.0d0) - t_0) * 3.0d0))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y);
	double t_1 = Math.sin(y) - (0.0625 * Math.sin(x));
	double t_2 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_3 = (2.0 + (t_1 * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (((t_2 * Math.cos(x)) - t_0) * 3.0));
	double tmp;
	if (x <= -2.3e+27) {
		tmp = t_3;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (t_1 * (((Math.sin(x) - (0.0625 * Math.sin(y))) * Math.sqrt(2.0)) * (1.0 - Math.cos(y))))) / (3.0 + (((t_2 * 1.0) - t_0) * 3.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y)
	t_1 = math.sin(y) - (0.0625 * math.sin(x))
	t_2 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_3 = (2.0 + (t_1 * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (((t_2 * math.cos(x)) - t_0) * 3.0))
	tmp = 0
	if x <= -2.3e+27:
		tmp = t_3
	elif x <= 1.45e-5:
		tmp = (2.0 + (t_1 * (((math.sin(x) - (0.0625 * math.sin(y))) * math.sqrt(2.0)) * (1.0 - math.cos(y))))) / (3.0 + (((t_2 * 1.0) - t_0) * 3.0))
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))
	t_1 = Float64(sin(y) - Float64(0.0625 * sin(x)))
	t_2 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_3 = Float64(Float64(2.0 + Float64(t_1 * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(t_2 * cos(x)) - t_0) * 3.0)))
	tmp = 0.0
	if (x <= -2.3e+27)
		tmp = t_3;
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sqrt(2.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 + Float64(Float64(Float64(t_2 * 1.0) - t_0) * 3.0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((sqrt(5.0) - 3.0) * 0.5) * cos(y);
	t_1 = sin(y) - (0.0625 * sin(x));
	t_2 = 0.5 * (sqrt(5.0) - 1.0);
	t_3 = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_2 * cos(x)) - t_0) * 3.0));
	tmp = 0.0;
	if (x <= -2.3e+27)
		tmp = t_3;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (t_1 * (((sin(x) - (0.0625 * sin(y))) * sqrt(2.0)) * (1.0 - cos(y))))) / (3.0 + (((t_2 * 1.0) - t_0) * 3.0));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e+27], t$95$3, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$2 * 1.0), $MachinePrecision] - t$95$0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e27 or 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

   if -2.3000000000000001e27 < x < 1.45e-5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.1%

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

Alternative 15: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_1 := \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(t\_0 \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_1
        (/
         (+
          2.0
          (*
           (- (sin y) (* 0.0625 (sin x)))
           (* (sin x) (* (sqrt 2.0) (- (cos x) 1.0)))))
         (+
          3.0
          (*
           (- (* t_0 (cos x)) (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
           3.0)))))
  (if (<= x -3.5)
    t_1
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
          (- (sin y) (* 0.0625 x)))
         (- (cos x) (cos y))))
       (* 3.0 (+ 1.0 (+ (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))) t_0))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	double tmp;
	if (x <= -3.5) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + ((0.5 * (cos(y) * (3.0 - sqrt(5.0)))) + t_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 = (2.0d0 + ((sin(y) - (0.0625d0 * sin(x))) * (sin(x) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (((t_0 * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    if (x <= (-3.5d0)) then
        tmp = t_1
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (0.0625d0 * sin(y)))) * (sin(y) - (0.0625d0 * x))) * (cos(x) - cos(y)))) / (3.0d0 * (1.0d0 + ((0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + t_0)))
    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 = (2.0 + ((Math.sin(y) - (0.0625 * Math.sin(x))) * (Math.sin(x) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (((t_0 * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	double tmp;
	if (x <= -3.5) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (0.0625 * Math.sin(y)))) * (Math.sin(y) - (0.0625 * x))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * (1.0 + ((0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_1 = (2.0 + ((math.sin(y) - (0.0625 * math.sin(x))) * (math.sin(x) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (((t_0 * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	tmp = 0
	if x <= -3.5:
		tmp = t_1
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (0.0625 * math.sin(y)))) * (math.sin(y) - (0.0625 * x))) * (math.cos(x) - math.cos(y)))) / (3.0 * (1.0 + ((0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + t_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(2.0 + Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * Float64(sin(x) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(t_0 * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)))
	tmp = 0.0
	if (x <= -3.5)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + t_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 = (2.0 + ((sin(y) - (0.0625 * sin(x))) * (sin(x) * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_0 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	tmp = 0.0;
	if (x <= -3.5)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + ((0.5 * (cos(y) * (3.0 - sqrt(5.0)))) + t_0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5], t$95$1, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

   if -3.5 < x < 1.45e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 60.1%

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.1%

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

Alternative 16: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot \left(t\_3 \cdot \sqrt{2}\right)\right) \cdot t\_0}{3 \cdot \left(\left(\left(0.5 \cdot t\_2\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot t\_0}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot t\_2\right) + t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(t\_1 \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) (cos y)))
       (t_1 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_2 (- 3.0 (sqrt 5.0)))
       (t_3 (pow (sin x) 2.0)))
  (if (<= x -3.5)
    (/
     (+ 2.0 (* (* -0.0625 (* t_3 (sqrt 2.0))) t_0))
     (*
      3.0
      (-
       (+ (* (* 0.5 t_2) (cos y)) 1.0)
       (* (* (- 1.0 (sqrt 5.0)) 0.5) (cos x)))))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
          (- (sin y) (* 0.0625 x)))
         t_0))
       (* 3.0 (+ 1.0 (+ (* 0.5 (* (cos y) t_2)) t_1))))
      (/
       (+ 2.0 (* -0.0625 (* t_3 (* (sqrt 2.0) (- (cos x) 1.0)))))
       (+
        3.0
        (*
         (-
          (* t_1 (cos x))
          (*
           (* (* (- 1.0 (/ 3.0 (sqrt 5.0))) (sqrt 5.0)) 0.5)
           (cos y)))
         3.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 0.5 * (sqrt(5.0) - 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -3.5) {
		tmp = (2.0 + ((-0.0625 * (t_3 * sqrt(2.0))) * t_0)) / (3.0 * ((((0.5 * t_2) * cos(y)) + 1.0) - (((1.0 - sqrt(5.0)) * 0.5) * cos(x))));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * t_0)) / (3.0 * (1.0 + ((0.5 * (cos(y) * t_2)) + t_1)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_1 * cos(x)) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = cos(x) - cos(y)
    t_1 = 0.5d0 * (sqrt(5.0d0) - 1.0d0)
    t_2 = 3.0d0 - sqrt(5.0d0)
    t_3 = sin(x) ** 2.0d0
    if (x <= (-3.5d0)) then
        tmp = (2.0d0 + (((-0.0625d0) * (t_3 * sqrt(2.0d0))) * t_0)) / (3.0d0 * ((((0.5d0 * t_2) * cos(y)) + 1.0d0) - (((1.0d0 - sqrt(5.0d0)) * 0.5d0) * cos(x))))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (0.0625d0 * sin(y)))) * (sin(y) - (0.0625d0 * x))) * t_0)) / (3.0d0 * (1.0d0 + ((0.5d0 * (cos(y) * t_2)) + t_1)))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (t_3 * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (3.0d0 + (((t_1 * cos(x)) - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = 0.5 * (Math.sqrt(5.0) - 1.0);
	double t_2 = 3.0 - Math.sqrt(5.0);
	double t_3 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -3.5) {
		tmp = (2.0 + ((-0.0625 * (t_3 * Math.sqrt(2.0))) * t_0)) / (3.0 * ((((0.5 * t_2) * Math.cos(y)) + 1.0) - (((1.0 - Math.sqrt(5.0)) * 0.5) * Math.cos(x))));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (0.0625 * Math.sin(y)))) * (Math.sin(y) - (0.0625 * x))) * t_0)) / (3.0 * (1.0 + ((0.5 * (Math.cos(y) * t_2)) + t_1)));
	} else {
		tmp = (2.0 + (-0.0625 * (t_3 * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (3.0 + (((t_1 * Math.cos(x)) - ((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y))) * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_2 = 3.0 - math.sqrt(5.0)
	t_3 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -3.5:
		tmp = (2.0 + ((-0.0625 * (t_3 * math.sqrt(2.0))) * t_0)) / (3.0 * ((((0.5 * t_2) * math.cos(y)) + 1.0) - (((1.0 - math.sqrt(5.0)) * 0.5) * math.cos(x))))
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (0.0625 * math.sin(y)))) * (math.sin(y) - (0.0625 * x))) * t_0)) / (3.0 * (1.0 + ((0.5 * (math.cos(y) * t_2)) + t_1)))
	else:
		tmp = (2.0 + (-0.0625 * (t_3 * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (3.0 + (((t_1 * math.cos(x)) - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 3.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * Float64(t_3 * sqrt(2.0))) * t_0)) / Float64(3.0 * Float64(Float64(Float64(Float64(0.5 * t_2) * cos(y)) + 1.0) - Float64(Float64(Float64(1.0 - sqrt(5.0)) * 0.5) * cos(x)))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(0.0625 * x))) * t_0)) / Float64(3.0 * Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * t_2)) + t_1))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_3 * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(3.0 + Float64(Float64(Float64(t_1 * cos(x)) - Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = 0.5 * (sqrt(5.0) - 1.0);
	t_2 = 3.0 - sqrt(5.0);
	t_3 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -3.5)
		tmp = (2.0 + ((-0.0625 * (t_3 * sqrt(2.0))) * t_0)) / (3.0 * ((((0.5 * t_2) * cos(y)) + 1.0) - (((1.0 - sqrt(5.0)) * 0.5) * cos(x))));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * t_0)) / (3.0 * (1.0 + ((0.5 * (cos(y) * t_2)) + t_1)));
	else
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * (cos(x) - 1.0))))) / (3.0 + (((t_1 * cos(x)) - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -3.5], N[(N[(2.0 + N[(N[(-0.0625 * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(1.0 - N[(3.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5

   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. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-+r- N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 61.7%

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

   6. Applied rewrites 61.7%

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

   if -3.5 < x < 1.45e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 60.1%

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.1%

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

   if 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

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

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

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

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

      5. lower-unsound-/.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

Alternative 17: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sqrt{5} - 1\right)\\ t_1 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\\ t_2 := t\_0 \cdot \cos x\\ \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{t\_1}{3 + \left(t\_2 - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + \left(t\_2 - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (- (sqrt 5.0) 1.0)))
       (t_1
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0))))))
       (t_2 (* t_0 (cos x))))
  (if (<= x -3.5)
    (/
     t_1
     (+ 3.0 (* (- t_2 (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y))) 3.0)))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
          (- (sin y) (* 0.0625 x)))
         (- (cos x) (cos y))))
       (* 3.0 (+ 1.0 (+ (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))) t_0))))
      (/
       t_1
       (+
        3.0
        (*
         (-
          t_2
          (*
           (* (* (- 1.0 (/ 3.0 (sqrt 5.0))) (sqrt 5.0)) 0.5)
           (cos y)))
         3.0)))))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (sqrt(5.0) - 1.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	double t_2 = t_0 * cos(x);
	double tmp;
	if (x <= -3.5) {
		tmp = t_1 / (3.0 + ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + ((0.5 * (cos(y) * (3.0 - sqrt(5.0)))) + t_0)));
	} else {
		tmp = t_1 / (3.0 + ((t_2 - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 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 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))
    t_2 = t_0 * cos(x)
    if (x <= (-3.5d0)) then
        tmp = t_1 / (3.0d0 + ((t_2 - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (0.0625d0 * sin(y)))) * (sin(y) - (0.0625d0 * x))) * (cos(x) - cos(y)))) / (3.0d0 * (1.0d0 + ((0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + t_0)))
    else
        tmp = t_1 / (3.0d0 + ((t_2 - ((((1.0d0 - (3.0d0 / sqrt(5.0d0))) * sqrt(5.0d0)) * 0.5d0) * cos(y))) * 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 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))));
	double t_2 = t_0 * Math.cos(x);
	double tmp;
	if (x <= -3.5) {
		tmp = t_1 / (3.0 + ((t_2 - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (0.0625 * Math.sin(y)))) * (Math.sin(y) - (0.0625 * x))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * (1.0 + ((0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + t_0)));
	} else {
		tmp = t_1 / (3.0 + ((t_2 - ((((1.0 - (3.0 / Math.sqrt(5.0))) * Math.sqrt(5.0)) * 0.5) * Math.cos(y))) * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.sqrt(5.0) - 1.0)
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))
	t_2 = t_0 * math.cos(x)
	tmp = 0
	if x <= -3.5:
		tmp = t_1 / (3.0 + ((t_2 - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (0.0625 * math.sin(y)))) * (math.sin(y) - (0.0625 * x))) * (math.cos(x) - math.cos(y)))) / (3.0 * (1.0 + ((0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + t_0)))
	else:
		tmp = t_1 / (3.0 + ((t_2 - ((((1.0 - (3.0 / math.sqrt(5.0))) * math.sqrt(5.0)) * 0.5) * math.cos(y))) * 3.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(sqrt(5.0) - 1.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0)))))
	t_2 = Float64(t_0 * cos(x))
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(t_2 - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + t_0))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(Float64(t_2 - Float64(Float64(Float64(Float64(1.0 - Float64(3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (sqrt(5.0) - 1.0);
	t_1 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))));
	t_2 = t_0 * cos(x);
	tmp = 0.0;
	if (x <= -3.5)
		tmp = t_1 / (3.0 + ((t_2 - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + ((0.5 * (cos(y) * (3.0 - sqrt(5.0)))) + t_0)));
	else
		tmp = t_1 / (3.0 + ((t_2 - ((((1.0 - (3.0 / sqrt(5.0))) * sqrt(5.0)) * 0.5) * cos(y))) * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5], N[(t$95$1 / N[(3.0 + N[(N[(t$95$2 - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(N[(t$95$2 - N[(N[(N[(N[(1.0 - N[(3.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 99.3%

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 61.7%

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

   8. Applied rewrites 61.7%

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

   if -3.5 < x < 1.45e-5

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 60.1%

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.1%

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 51.1%

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

   10. Applied rewrites 51.1%

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

   if 1.45e-5 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      8. lower-cos.f64 61.7%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   8. Applied rewrites 61.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ t_1 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := 0.5 \cdot t\_2\\ t_4 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_4 \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + \left(t\_3 \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot x\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(t\_1 + t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(t\_4 \cdot t\_0\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) 1.0))
       (t_1 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3 (* 0.5 t_2))
       (t_4 (pow (sin x) 2.0)))
  (if (<= x -3.5)
    (/
     (+ 2.0 (* -0.0625 (* t_4 (* (sqrt 2.0) t_0))))
     (+
      3.0
      (*
       (- (* t_3 (cos x)) (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
       3.0)))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (*
          (* (sqrt 2.0) (- x (* 0.0625 (sin y))))
          (- (sin y) (* 0.0625 x)))
         (- (cos x) (cos y))))
       (* 3.0 (+ 1.0 (+ t_1 t_3))))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* t_4 t_0))))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_2)) t_1))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 0.5 * t_2;
	double t_4 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -3.5) {
		tmp = (2.0 + (-0.0625 * (t_4 * (sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (cos(x) * t_2)) + 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos(x) - 1.0d0
    t_1 = 0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = 0.5d0 * t_2
    t_4 = sin(x) ** 2.0d0
    if (x <= (-3.5d0)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_4 * (sqrt(2.0d0) * t_0)))) / (3.0d0 + (((t_3 * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (0.0625d0 * sin(y)))) * (sin(y) - (0.0625d0 * x))) * (cos(x) - cos(y)))) / (3.0d0 * (1.0d0 + (t_1 + t_3)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * (t_4 * t_0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - 1.0;
	double t_1 = 0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)));
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = 0.5 * t_2;
	double t_4 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -3.5) {
		tmp = (2.0 + (-0.0625 * (t_4 * (Math.sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (0.0625 * Math.sin(y)))) * (Math.sin(y) - (0.0625 * x))) * (Math.cos(x) - Math.cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - 1.0
	t_1 = 0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = 0.5 * t_2
	t_4 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -3.5:
		tmp = (2.0 + (-0.0625 * (t_4 * (math.sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (0.0625 * math.sin(y)))) * (math.sin(y) - (0.0625 * x))) * (math.cos(x) - math.cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(0.5 * t_2)
	t_4 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -3.5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_4 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(Float64(Float64(t_3 * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * Float64(sin(y) - Float64(0.0625 * x))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(t_1 + t_3))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(t_4 * t_0)))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - 1.0;
	t_1 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	t_2 = sqrt(5.0) - 1.0;
	t_3 = 0.5 * t_2;
	t_4 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -3.5)
		tmp = (2.0 + (-0.0625 * (t_4 * (sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * (sin(y) - (0.0625 * x))) * (cos(x) - cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	else
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (cos(x) * t_2)) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -3.5], N[(N[(2.0 + N[(-0.0625 * N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{\sin x}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      7. mult-flip N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\sin x \cdot \frac{1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y - \sin x \cdot \color{blue}{\frac{1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{1}{16} \cdot \sin x}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{1}{16} \cdot \sin x}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      11. lower-*.f64 99.3%

          \[\leadsto \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \color{blue}{\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      8. lower-cos.f64 61.7%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   8. Applied rewrites 61.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   if -3.5 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \color{blue}{\frac{1}{16} \cdot \sin y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{1}{16} \cdot \color{blue}{\sin y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-sin.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \color{blue}{\frac{1}{16} \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - 0.0625 \cdot \color{blue}{x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   10. Applied rewrites 51.1%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \color{blue}{0.0625 \cdot x}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   if 1.45e-5 < 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 x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-cos.f64 61.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ t_1 := 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\\ t_2 := \sqrt{5} - 1\\ t_3 := 0.5 \cdot t\_2\\ t_4 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1550:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_4 \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + \left(t\_3 \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(t\_1 + t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left(t\_4 \cdot t\_0\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) 1.0))
       (t_1 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0)))))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3 (* 0.5 t_2))
       (t_4 (pow (sin x) 2.0)))
  (if (<= x -1550.0)
    (/
     (+ 2.0 (* -0.0625 (* t_4 (* (sqrt 2.0) t_0))))
     (+
      3.0
      (*
       (- (* t_3 (cos x)) (* (* (- (sqrt 5.0) 3.0) 0.5) (cos y)))
       3.0)))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         (* (* (sqrt 2.0) (- x (* 0.0625 (sin y)))) (sin y))
         (- (cos x) (cos y))))
       (* 3.0 (+ 1.0 (+ t_1 t_3))))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (sqrt 2.0) (* -0.0625 (* t_4 t_0))))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_2)) t_1))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 0.5 * t_2;
	double t_4 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -1550.0) {
		tmp = (2.0 + (-0.0625 * (t_4 * (sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (cos(x) * t_2)) + 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos(x) - 1.0d0
    t_1 = 0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = 0.5d0 * t_2
    t_4 = sin(x) ** 2.0d0
    if (x <= (-1550.0d0)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_4 * (sqrt(2.0d0) * t_0)))) / (3.0d0 + (((t_3 * cos(x)) - (((sqrt(5.0d0) - 3.0d0) * 0.5d0) * cos(y))) * 3.0d0))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (x - (0.0625d0 * sin(y)))) * sin(y)) * (cos(x) - cos(y)))) / (3.0d0 * (1.0d0 + (t_1 + t_3)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * (t_4 * t_0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + t_1)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - 1.0;
	double t_1 = 0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)));
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = 0.5 * t_2;
	double t_4 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -1550.0) {
		tmp = (2.0 + (-0.0625 * (t_4 * (Math.sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * Math.cos(x)) - (((Math.sqrt(5.0) - 3.0) * 0.5) * Math.cos(y))) * 3.0));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (x - (0.0625 * Math.sin(y)))) * Math.sin(y)) * (Math.cos(x) - Math.cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - 1.0
	t_1 = 0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = 0.5 * t_2
	t_4 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -1550.0:
		tmp = (2.0 + (-0.0625 * (t_4 * (math.sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * math.cos(x)) - (((math.sqrt(5.0) - 3.0) * 0.5) * math.cos(y))) * 3.0))
	elif x <= 1.45e-5:
		tmp = (2.0 + (((math.sqrt(2.0) * (x - (0.0625 * math.sin(y)))) * math.sin(y)) * (math.cos(x) - math.cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(0.5 * t_2)
	t_4 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -1550.0)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_4 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(Float64(Float64(t_3 * cos(x)) - Float64(Float64(Float64(sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0)));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(x - Float64(0.0625 * sin(y)))) * sin(y)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(1.0 + Float64(t_1 + t_3))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64(t_4 * t_0)))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - 1.0;
	t_1 = 0.5 * (cos(y) * (3.0 - sqrt(5.0)));
	t_2 = sqrt(5.0) - 1.0;
	t_3 = 0.5 * t_2;
	t_4 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -1550.0)
		tmp = (2.0 + (-0.0625 * (t_4 * (sqrt(2.0) * t_0)))) / (3.0 + (((t_3 * cos(x)) - (((sqrt(5.0) - 3.0) * 0.5) * cos(y))) * 3.0));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (((sqrt(2.0) * (x - (0.0625 * sin(y)))) * sin(y)) * (cos(x) - cos(y)))) / (3.0 * (1.0 + (t_1 + t_3)));
	else
		tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (t_4 * t_0)))) / (1.0 + ((0.5 * (cos(x) * t_2)) + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -1550.0], N[(N[(2.0 + N[(-0.0625 * N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(x - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$1 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1550

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{\sin x}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      7. mult-flip N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\sin x \cdot \frac{1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sin y - \sin x \cdot \color{blue}{\frac{1}{16}}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{1}{16} \cdot \sin x}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\sin y - \color{blue}{\frac{1}{16} \cdot \sin x}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      11. lower-*.f64 99.3%

          \[\leadsto \frac{2 + \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \color{blue}{\left(\sin y - 0.0625 \cdot \sin x\right) \cdot \left(\left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x - 1\right)}\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\cos x} - 1\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - \color{blue}{1}\right)\right)\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      8. lower-cos.f64 61.7%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   8. Applied rewrites 61.7%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   if -1550 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - \frac{1}{16} \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \color{blue}{\frac{1}{16} \cdot \sin y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - \frac{1}{16} \cdot \color{blue}{\sin y}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-sin.f64 51.1%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 51.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(x - 0.0625 \cdot \sin y\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-sin.f64 53.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   10. Applied rewrites 53.3%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(x - 0.0625 \cdot \sin y\right)\right) \cdot \color{blue}{\sin y}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   if 1.45e-5 < 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 x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-cos.f64 61.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\\ t_2 := 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\\ \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(1 + \left(t\_2 + 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)\right)}\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(t\_2 + 0.38196601125010515\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\left(\left(0.5 \cdot t\_0\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 3.0 (sqrt 5.0)))
       (t_1
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y)))))))
       (t_2 (* 0.5 (* (cos x) (- (sqrt 5.0) 1.0)))))
  (if (<= y -19.5)
    (/ t_1 (* 3.0 (+ 1.0 (+ t_2 (* 0.5 (* (cos y) t_0))))))
    (if (<= y 2.16e-7)
      (*
       0.3333333333333333
       (/
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ 1.0 (+ t_2 0.38196601125010515))))
      (/
       t_1
       (*
        3.0
        (-
         (+ (* (* 0.5 t_0) (cos y)) 1.0)
         (* (* (- 1.0 (sqrt 5.0)) 0.5) (cos x)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))));
	double t_2 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1 / (3.0 * (1.0 + (t_2 + (0.5 * (cos(y) * t_0)))));
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_2 + 0.38196601125010515)));
	} else {
		tmp = t_1 / (3.0 * ((((0.5 * t_0) * cos(y)) + 1.0) - (((1.0 - sqrt(5.0)) * 0.5) * cos(x))));
	}
	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 = 3.0d0 - sqrt(5.0d0)
    t_1 = 2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))
    t_2 = 0.5d0 * (cos(x) * (sqrt(5.0d0) - 1.0d0))
    if (y <= (-19.5d0)) then
        tmp = t_1 / (3.0d0 * (1.0d0 + (t_2 + (0.5d0 * (cos(y) * t_0)))))
    else if (y <= 2.16d-7) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.0d0 + (t_2 + 0.38196601125010515d0)))
    else
        tmp = t_1 / (3.0d0 * ((((0.5d0 * t_0) * cos(y)) + 1.0d0) - (((1.0d0 - sqrt(5.0d0)) * 0.5d0) * cos(x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = 2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))));
	double t_2 = 0.5 * (Math.cos(x) * (Math.sqrt(5.0) - 1.0));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1 / (3.0 * (1.0 + (t_2 + (0.5 * (Math.cos(y) * t_0)))));
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.0 + (t_2 + 0.38196601125010515)));
	} else {
		tmp = t_1 / (3.0 * ((((0.5 * t_0) * Math.cos(y)) + 1.0) - (((1.0 - Math.sqrt(5.0)) * 0.5) * Math.cos(x))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = 2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))
	t_2 = 0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))
	tmp = 0
	if y <= -19.5:
		tmp = t_1 / (3.0 * (1.0 + (t_2 + (0.5 * (math.cos(y) * t_0)))))
	elif y <= 2.16e-7:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.0 + (t_2 + 0.38196601125010515)))
	else:
		tmp = t_1 / (3.0 * ((((0.5 * t_0) * math.cos(y)) + 1.0) - (((1.0 - math.sqrt(5.0)) * 0.5) * math.cos(x))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y))))))
	t_2 = Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0)))
	tmp = 0.0
	if (y <= -19.5)
		tmp = Float64(t_1 / Float64(3.0 * Float64(1.0 + Float64(t_2 + Float64(0.5 * Float64(cos(y) * t_0))))));
	elseif (y <= 2.16e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + Float64(t_2 + 0.38196601125010515))));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(Float64(Float64(0.5 * t_0) * cos(y)) + 1.0) - Float64(Float64(Float64(1.0 - sqrt(5.0)) * 0.5) * cos(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = 2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))));
	t_2 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	tmp = 0.0;
	if (y <= -19.5)
		tmp = t_1 / (3.0 * (1.0 + (t_2 + (0.5 * (cos(y) * t_0)))));
	elseif (y <= 2.16e-7)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_2 + 0.38196601125010515)));
	else
		tmp = t_1 / (3.0 * ((((0.5 * t_0) * cos(y)) + 1.0) - (((1.0 - sqrt(5.0)) * 0.5) * cos(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -19.5], N[(t$95$1 / N[(3.0 * N[(1.0 + N[(t$95$2 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.16e-7], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 + 0.38196601125010515), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(N[(1.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -19.5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(\color{blue}{3} - \sqrt{5}\right)\right)\right)\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)\right)\right)} \]

      12. lower-sqrt.f64 62.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   10. Applied rewrites 62.5%

       \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   if -19.5 < y < 2.16e-7

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   5. Evaluated real constant 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.38196601125010515\right)} \]

   if 2.16e-7 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   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. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)\right)} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \color{blue}{\left(1 - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)}\right)} \]

      6. associate-+r- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)}} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right)} - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      10. mult-flip N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y + 1\right) - \left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{5} - 1}{2}\right)\right) \cdot \cos x}\right)} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)} \]

      8. lower-cos.f64 62.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)} \]

   6. Applied rewrites 62.5%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(\left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) - \left(\left(1 - \sqrt{5}\right) \cdot 0.5\right) \cdot \cos x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\\ t_1 := \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(t\_0 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}\\ \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(t\_0 + 0.38196601125010515\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (* (cos x) (- (sqrt 5.0) 1.0))))
       (t_1
        (/
         (+
          2.0
          (*
           -0.0625
           (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (*
          3.0
          (+ 1.0 (+ t_0 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))))
  (if (<= y -19.5)
    t_1
    (if (<= y 2.16e-7)
      (*
       0.3333333333333333
       (/
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ 1.0 (+ t_0 0.38196601125010515))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	double t_1 = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + (t_0 + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	} 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 * (cos(x) * (sqrt(5.0d0) - 1.0d0))
    t_1 = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * (1.0d0 + (t_0 + (0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    if (y <= (-19.5d0)) then
        tmp = t_1
    else if (y <= 2.16d-7) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.0d0 + (t_0 + 0.38196601125010515d0)))
    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.cos(x) * (Math.sqrt(5.0) - 1.0));
	double t_1 = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * (1.0 + (t_0 + (0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))
	t_1 = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * (1.0 + (t_0 + (0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	tmp = 0
	if y <= -19.5:
		tmp = t_1
	elif y <= 2.16e-7:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0)))
	t_1 = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + Float64(t_0 + Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))))
	tmp = 0.0
	if (y <= -19.5)
		tmp = t_1;
	elseif (y <= 2.16e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + Float64(t_0 + 0.38196601125010515))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	t_1 = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + (t_0 + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	tmp = 0.0;
	if (y <= -19.5)
		tmp = t_1;
	elseif (y <= 2.16e-7)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(t$95$0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -19.5], t$95$1, If[LessEqual[y, 2.16e-7], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 + 0.38196601125010515), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -19.5 or 2.16e-7 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right)\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(\color{blue}{3} - \sqrt{5}\right)\right)\right)\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right)\right)\right)\right)} \]

      12. lower-sqrt.f64 62.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)} \]

   10. Applied rewrites 62.5%

       \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   if -19.5 < y < 2.16e-7

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   5. Evaluated real constant 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.38196601125010515\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\\ t_1 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(t\_0 + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(t\_0 + 0.38196601125010515\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* 0.5 (* (cos x) (- (sqrt 5.0) 1.0))))
       (t_1
        (*
         0.3333333333333333
         (/
          (+
           2.0
           (*
            (sqrt 2.0)
            (* -0.0625 (* (pow (sin y) 2.0) (- 1.0 (cos y))))))
          (+ 1.0 (+ t_0 (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))))
  (if (<= y -19.5)
    t_1
    (if (<= y 2.16e-7)
      (*
       0.3333333333333333
       (/
        (+
         2.0
         (*
          -0.0625
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (- (cos x) 1.0)))))
        (+ 1.0 (+ t_0 0.38196601125010515))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	double t_1 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(y), 2.0) * (1.0 - cos(y)))))) / (1.0 + (t_0 + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	} 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 * (cos(x) * (sqrt(5.0d0) - 1.0d0))
    t_1 = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((sin(y) ** 2.0d0) * (1.0d0 - cos(y)))))) / (1.0d0 + (t_0 + (0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    if (y <= (-19.5d0)) then
        tmp = t_1
    else if (y <= 2.16d-7) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) - 1.0d0))))) / (1.0d0 + (t_0 + 0.38196601125010515d0)))
    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.cos(x) * (Math.sqrt(5.0) - 1.0));
	double t_1 = 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (1.0 - Math.cos(y)))))) / (1.0 + (t_0 + (0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	double tmp;
	if (y <= -19.5) {
		tmp = t_1;
	} else if (y <= 2.16e-7) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (math.cos(x) * (math.sqrt(5.0) - 1.0))
	t_1 = 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * (-0.0625 * (math.pow(math.sin(y), 2.0) * (1.0 - math.cos(y)))))) / (1.0 + (t_0 + (0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	tmp = 0
	if y <= -19.5:
		tmp = t_1
	elif y <= 2.16e-7:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(cos(x) * Float64(sqrt(5.0) - 1.0)))
	t_1 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(1.0 - cos(y)))))) / Float64(1.0 + Float64(t_0 + Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))))
	tmp = 0.0
	if (y <= -19.5)
		tmp = t_1;
	elseif (y <= 2.16e-7)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + Float64(t_0 + 0.38196601125010515))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (cos(x) * (sqrt(5.0) - 1.0));
	t_1 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * ((sin(y) ^ 2.0) * (1.0 - cos(y)))))) / (1.0 + (t_0 + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	tmp = 0.0;
	if (y <= -19.5)
		tmp = t_1;
	elseif (y <= 2.16e-7)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) - 1.0))))) / (1.0 + (t_0 + 0.38196601125010515)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -19.5], t$95$1, If[LessEqual[y, 2.16e-7], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 + 0.38196601125010515), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -19.5 or 2.16e-7 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-cos.f64 62.4%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 62.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -19.5 < y < 2.16e-7

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   5. Evaluated real constant 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.38196601125010515\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 78.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_0\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot t\_0 - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1
        (*
         0.3333333333333333
         (/
          (+
           2.0
           (*
            (sqrt 2.0)
            (* -0.0625 (* (pow (sin x) 2.0) (- (cos x) 1.0)))))
          (+
           1.0
           (+
            (* 0.5 (* (cos x) t_0))
            (* 0.5 (* (cos y) (- 3.0 (sqrt 5.0))))))))))
  (if (<= x -0.024)
    t_1
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         3.0
         (-
          (* 0.5 t_0)
          (*
           0.5
           (*
            (cos y)
            (* (sqrt 5.0) (- 1.0 (* 3.0 (/ 1.0 (sqrt 5.0)))))))))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * (pow(sin(x), 2.0) * (cos(x) - 1.0))))) / (1.0 + ((0.5 * (cos(x) * t_0)) + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	double tmp;
	if (x <= -0.024) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_0) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	} 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 = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((-0.0625d0) * ((sin(x) ** 2.0d0) * (cos(x) - 1.0d0))))) / (1.0d0 + ((0.5d0 * (cos(x) * t_0)) + (0.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    if (x <= (-0.024d0)) then
        tmp = t_1
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((0.5d0 * t_0) - (0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))))))
    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 = 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.cos(x) - 1.0))))) / (1.0 + ((0.5 * (Math.cos(x) * t_0)) + (0.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	double tmp;
	if (x <= -0.024) {
		tmp = t_1;
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_0) - (0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.cos(x) - 1.0))))) / (1.0 + ((0.5 * (math.cos(x) * t_0)) + (0.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	tmp = 0
	if x <= -0.024:
		tmp = t_1
	elif x <= 1.45e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_0) - (0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) - 1.0))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_0)) + Float64(0.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))))
	tmp = 0.0
	if (x <= -0.024)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(0.5 * t_0) - Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * (-0.0625 * ((sin(x) ^ 2.0) * (cos(x) - 1.0))))) / (1.0 + ((0.5 * (cos(x) * t_0)) + (0.5 * (cos(y) * (3.0 - sqrt(5.0)))))));
	tmp = 0.0;
	if (x <= -0.024)
		tmp = t_1;
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_0) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.024], t$95$1, If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(0.5 * t$95$0), $MachinePrecision] - N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * N[(1.0 - N[(3.0 * N[(1.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.024 or 1.45e-5 < 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 x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-cos.f64 61.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 61.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -0.024 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]

   8. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 77.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\cos x - 1\right) \cdot \sqrt{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := \left(t\_1 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{t\_2}{\left(-0.0625 \cdot t\_0\right) \cdot t\_3 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot t\_1 - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{2}{t\_2} - \frac{\left(0.0625 \cdot t\_0\right) \cdot t\_3}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (- (cos x) 1.0) (sqrt 2.0)))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2 (- (* (- (* t_1 (cos x)) (- (sqrt 5.0) 3.0)) 0.5) -1.0))
       (t_3 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/ 1.0 (/ t_2 (- (* (* -0.0625 t_0) t_3) -2.0))))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         3.0
         (-
          (* 0.5 t_1)
          (*
           0.5
           (*
            (cos y)
            (* (sqrt 5.0) (- 1.0 (* 3.0 (/ 1.0 (sqrt 5.0)))))))))))
      (*
       0.3333333333333333
       (- (/ 2.0 t_2) (/ (* (* 0.0625 t_0) t_3) t_2)))))))

C

double code(double x, double y) {
	double t_0 = (cos(x) - 1.0) * sqrt(2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = (((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0;
	double t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (t_2 / (((-0.0625 * t_0) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * t_0) * t_3) / 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) :: t_3
    real(8) :: tmp
    t_0 = (cos(x) - 1.0d0) * sqrt(2.0d0)
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = (((t_1 * cos(x)) - (sqrt(5.0d0) - 3.0d0)) * 0.5d0) - (-1.0d0)
    t_3 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (t_2 / ((((-0.0625d0) * t_0) * t_3) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((0.5d0 * t_1) - (0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 / t_2) - (((0.0625d0 * t_0) * t_3) / t_2))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.cos(x) - 1.0) * Math.sqrt(2.0);
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = (((t_1 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0)) * 0.5) - -1.0;
	double t_3 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (t_2 / (((-0.0625 * t_0) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * t_0) * t_3) / t_2));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.cos(x) - 1.0) * math.sqrt(2.0)
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = (((t_1 * math.cos(x)) - (math.sqrt(5.0) - 3.0)) * 0.5) - -1.0
	t_3 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (t_2 / (((-0.0625 * t_0) * t_3) - -2.0)))
	elif x <= 1.45e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))))))
	else:
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * t_0) * t_3) / t_2))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(cos(x) - 1.0) * sqrt(2.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(Float64(Float64(Float64(t_1 * cos(x)) - Float64(sqrt(5.0) - 3.0)) * 0.5) - -1.0)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(t_2 / Float64(Float64(Float64(-0.0625 * t_0) * t_3) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(0.5 * t_1) - Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 / t_2) - Float64(Float64(Float64(0.0625 * t_0) * t_3) / t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (cos(x) - 1.0) * sqrt(2.0);
	t_1 = sqrt(5.0) - 1.0;
	t_2 = (((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0;
	t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (t_2 / (((-0.0625 * t_0) * t_3) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	else
		tmp = 0.3333333333333333 * ((2.0 / t_2) - (((0.0625 * t_0) * t_3) / t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(t$95$2 / N[(N[(N[(-0.0625 * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(0.5 * t$95$1), $MachinePrecision] - N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * N[(1.0 - N[(3.0 * N[(1.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 / t$95$2), $MachinePrecision] - N[(N[(N[(0.0625 * t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]

   8. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{2}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1} - \color{blue}{\frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right)}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 77.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\left(t\_1 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right)\right) \cdot t\_2 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot t\_1 - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot \left(t\_0 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) 1.0))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/
      1.0
      (/
       (- (* (- (* t_1 (cos x)) (- (sqrt 5.0) 3.0)) 0.5) -1.0)
       (- (* (* -0.0625 (* t_0 (sqrt 2.0))) t_2) -2.0))))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        3.0
        (*
         3.0
         (-
          (* 0.5 t_1)
          (*
           0.5
           (*
            (cos y)
            (* (sqrt 5.0) (- 1.0 (* 3.0 (/ 1.0 (sqrt 5.0)))))))))))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_2 (sqrt 2.0)) (* t_0 -0.0625)))
        (+
         1.0
         (+ (* 0.5 (* (cos x) t_1)) (* 0.5 (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_2) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_1)) + (0.5 * (3.0 - sqrt(5.0))))));
	}
	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 = cos(x) - 1.0d0
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (((((t_1 * cos(x)) - (sqrt(5.0d0) - 3.0d0)) * 0.5d0) - (-1.0d0)) / ((((-0.0625d0) * (t_0 * sqrt(2.0d0))) * t_2) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((0.5d0 * t_1) - (0.5d0 * (cos(y) * (sqrt(5.0d0) * (1.0d0 - (3.0d0 * (1.0d0 / sqrt(5.0d0))))))))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_2 * sqrt(2.0d0)) * (t_0 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_1)) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - 1.0;
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * Math.sqrt(2.0))) * t_2) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (Math.cos(y) * (Math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / Math.sqrt(5.0))))))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * Math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_1)) + (0.5 * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - 1.0
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * math.cos(x)) - (math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * math.sqrt(2.0))) * t_2) - -2.0)))
	elif x <= 1.45e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (math.cos(y) * (math.sqrt(5.0) * (1.0 - (3.0 * (1.0 / math.sqrt(5.0))))))))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_1)) + (0.5 * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_1 * cos(x)) - Float64(sqrt(5.0) - 3.0)) * 0.5) - -1.0) / Float64(Float64(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))) * t_2) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(0.5 * t_1) - Float64(0.5 * Float64(cos(y) * Float64(sqrt(5.0) * Float64(1.0 - Float64(3.0 * Float64(1.0 / sqrt(5.0)))))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * Float64(t_0 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_1)) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - 1.0;
	t_1 = sqrt(5.0) - 1.0;
	t_2 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_2) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_1) - (0.5 * (cos(y) * (sqrt(5.0) * (1.0 - (3.0 * (1.0 / sqrt(5.0))))))))));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_1)) + (0.5 * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(0.5 * t$95$1), $MachinePrecision] - N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * N[(1.0 - N[(3.0 * N[(1.0 / N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\sqrt{5} - 3\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      4. lower-unsound--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\color{blue}{\left(1 - \frac{3}{\sqrt{5}}\right)} \cdot \sqrt{5}\right) \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3} \]

      5. lower-unsound-/.f64 99.3%

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\left(1 - \color{blue}{\frac{3}{\sqrt{5}}}\right) \cdot \sqrt{5}\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\color{blue}{\left(\left(1 - \frac{3}{\sqrt{5}}\right) \cdot \sqrt{5}\right)} \cdot 0.5\right) \cdot \cos y\right) \cdot 3} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]

   8. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} \cdot \left(1 - 3 \cdot \frac{1}{\sqrt{5}}\right)\right)\right)\right)}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 3\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\left(t\_2 \cdot \cos x - t\_0\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot t\_3 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot t\_2 - 0.5 \cdot \left(\cos y \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_3 \cdot \sqrt{2}\right) \cdot \left(t\_1 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 3.0))
       (t_1 (- (cos x) 1.0))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/
      1.0
      (/
       (- (* (- (* t_2 (cos x)) t_0) 0.5) -1.0)
       (- (* (* -0.0625 (* t_1 (sqrt 2.0))) t_3) -2.0))))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 3.0 (* 3.0 (- (* 0.5 t_2) (* 0.5 (* (cos y) t_0))))))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_3 (sqrt 2.0)) (* t_1 -0.0625)))
        (+
         1.0
         (+ (* 0.5 (* (cos x) t_2)) (* 0.5 (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 3.0;
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - t_0) * 0.5) - -1.0) / (((-0.0625 * (t_1 * sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_2) - (0.5 * (cos(y) * t_0)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * (3.0 - sqrt(5.0))))));
	}
	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) - 3.0d0
    t_1 = cos(x) - 1.0d0
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (((((t_2 * cos(x)) - t_0) * 0.5d0) - (-1.0d0)) / ((((-0.0625d0) * (t_1 * sqrt(2.0d0))) * t_3) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((0.5d0 * t_2) - (0.5d0 * (cos(y) * t_0)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_3 * sqrt(2.0d0)) * (t_1 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    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.cos(x) - 1.0;
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * Math.cos(x)) - t_0) * 0.5) - -1.0) / (((-0.0625 * (t_1 * Math.sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_2) - (0.5 * (Math.cos(y) * t_0)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * Math.sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + (0.5 * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 3.0
	t_1 = math.cos(x) - 1.0
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * math.cos(x)) - t_0) * 0.5) - -1.0) / (((-0.0625 * (t_1 * math.sqrt(2.0))) * t_3) - -2.0)))
	elif x <= 1.45e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_2) - (0.5 * (math.cos(y) * t_0)))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * math.sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + (0.5 * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 3.0)
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_2 * cos(x)) - t_0) * 0.5) - -1.0) / Float64(Float64(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))) * t_3) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(0.5 * t_2) - Float64(0.5 * Float64(cos(y) * t_0))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_3 * sqrt(2.0)) * Float64(t_1 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 3.0;
	t_1 = cos(x) - 1.0;
	t_2 = sqrt(5.0) - 1.0;
	t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - t_0) * 0.5) - -1.0) / (((-0.0625 * (t_1 * sqrt(2.0))) * t_3) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((0.5 * t_2) - (0.5 * (cos(y) * t_0)))));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(0.5 * t$95$2), $MachinePrecision] - N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-+l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   3. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\left(0.5 \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x - \left(\left(\sqrt{5} - 3\right) \cdot 0.5\right) \cdot \cos y\right) \cdot 3}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\color{blue}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) - \frac{1}{2} \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}} \]

   6. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \left(\cos y \cdot \left(\sqrt{5} - 3\right)\right)\right)}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ t_1 := \sqrt{5} - 1\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\left(t\_1 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right)\right) \cdot t\_2 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot 0.7639320225002103\right) + 0.5 \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot \left(t\_0 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_1\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) 1.0))
       (t_1 (- (sqrt 5.0) 1.0))
       (t_2 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/
      1.0
      (/
       (- (* (- (* t_1 (cos x)) (- (sqrt 5.0) 3.0)) 0.5) -1.0)
       (- (* (* -0.0625 (* t_0 (sqrt 2.0))) t_2) -2.0))))
    (if (<= x 1.45e-5)
      (/
       (+
        2.0
        (*
         -0.0625
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (*
        3.0
        (+
         1.0
         (+ (* 0.5 (* (cos y) 0.7639320225002103)) (* 0.5 t_1)))))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_2 (sqrt 2.0)) (* t_0 -0.0625)))
        (+
         1.0
         (+ (* 0.5 (* (cos x) t_1)) (* 0.5 (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_2) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + ((0.5 * (cos(y) * 0.7639320225002103)) + (0.5 * t_1))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_1)) + (0.5 * (3.0 - sqrt(5.0))))));
	}
	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 = cos(x) - 1.0d0
    t_1 = sqrt(5.0d0) - 1.0d0
    t_2 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (((((t_1 * cos(x)) - (sqrt(5.0d0) - 3.0d0)) * 0.5d0) - (-1.0d0)) / ((((-0.0625d0) * (t_0 * sqrt(2.0d0))) * t_2) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * (1.0d0 + ((0.5d0 * (cos(y) * 0.7639320225002103d0)) + (0.5d0 * t_1))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_2 * sqrt(2.0d0)) * (t_0 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_1)) + (0.5d0 * (3.0d0 - sqrt(5.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - 1.0;
	double t_1 = Math.sqrt(5.0) - 1.0;
	double t_2 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * Math.sqrt(2.0))) * t_2) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * (1.0 + ((0.5 * (Math.cos(y) * 0.7639320225002103)) + (0.5 * t_1))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * Math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_1)) + (0.5 * (3.0 - Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - 1.0
	t_1 = math.sqrt(5.0) - 1.0
	t_2 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * math.cos(x)) - (math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * math.sqrt(2.0))) * t_2) - -2.0)))
	elif x <= 1.45e-5:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * (1.0 + ((0.5 * (math.cos(y) * 0.7639320225002103)) + (0.5 * t_1))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_1)) + (0.5 * (3.0 - math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_1 * cos(x)) - Float64(sqrt(5.0) - 3.0)) * 0.5) - -1.0) / Float64(Float64(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))) * t_2) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * 0.7639320225002103)) + Float64(0.5 * t_1)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * Float64(t_0 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_1)) + Float64(0.5 * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - 1.0;
	t_1 = sqrt(5.0) - 1.0;
	t_2 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (((((t_1 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_2) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (1.0 + ((0.5 * (cos(y) * 0.7639320225002103)) + (0.5 * t_1))));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_2 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_1)) + (0.5 * (3.0 - sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[(N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * 0.7639320225002103), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   8. Evaluated real constant 59.6%

      \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot 0.7639320225002103\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 77.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \cos x - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\left(t\_2 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(t\_0 \cdot \sqrt{2}\right)\right) \cdot t\_3 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot \left(t\_2 + t\_1 \cdot \cos y\right) - -1\right) \cdot 3}{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_3 \cdot \sqrt{2}\right) \cdot \left(t\_0 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + 0.5 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (cos x) 1.0))
       (t_1 (- 3.0 (sqrt 5.0)))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/
      1.0
      (/
       (- (* (- (* t_2 (cos x)) (- (sqrt 5.0) 3.0)) 0.5) -1.0)
       (- (* (* -0.0625 (* t_0 (sqrt 2.0))) t_3) -2.0))))
    (if (<= x 1.45e-5)
      (/
       1.0
       (/
        (* (- (* 0.5 (+ t_2 (* t_1 (cos y)))) -1.0) 3.0)
        (-
         (*
          (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         -2.0)))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_3 (sqrt 2.0)) (* t_0 -0.0625)))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_2)) (* 0.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = 1.0 / ((((0.5 * (t_2 + (t_1 * cos(y)))) - -1.0) * 3.0) / (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(x) - 1.0d0
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (((((t_2 * cos(x)) - (sqrt(5.0d0) - 3.0d0)) * 0.5d0) - (-1.0d0)) / ((((-0.0625d0) * (t_0 * sqrt(2.0d0))) * t_3) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = 1.0d0 / ((((0.5d0 * (t_2 + (t_1 * cos(y)))) - (-1.0d0)) * 3.0d0) / ((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_3 * sqrt(2.0d0)) * (t_0 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + (0.5d0 * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - 1.0;
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * Math.sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = 1.0 / ((((0.5 * (t_2 + (t_1 * Math.cos(y)))) - -1.0) * 3.0) / (((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * Math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + (0.5 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - 1.0
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * math.cos(x)) - (math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * math.sqrt(2.0))) * t_3) - -2.0)))
	elif x <= 1.45e-5:
		tmp = 1.0 / ((((0.5 * (t_2 + (t_1 * math.cos(y)))) - -1.0) * 3.0) / (((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * math.sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + (0.5 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_2 * cos(x)) - Float64(sqrt(5.0) - 3.0)) * 0.5) - -1.0) / Float64(Float64(Float64(-0.0625 * Float64(t_0 * sqrt(2.0))) * t_3) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(0.5 * Float64(t_2 + Float64(t_1 * cos(y)))) - -1.0) * 3.0) / Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_3 * sqrt(2.0)) * Float64(t_0 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + Float64(0.5 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - 1.0;
	t_1 = 3.0 - sqrt(5.0);
	t_2 = sqrt(5.0) - 1.0;
	t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_0 * sqrt(2.0))) * t_3) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = 1.0 / ((((0.5 * (t_2 + (t_1 * cos(y)))) - -1.0) * 3.0) / (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_0 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(1.0 / N[(N[(N[(N[(0.5 * N[(t$95$2 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * 3.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1\right) \cdot 3}{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 77.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - 1\\ t_2 := \sqrt{5} - 1\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{\left(t\_2 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(t\_1 \cdot \sqrt{2}\right)\right) \cdot t\_3 - -2}}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{0.5 \cdot \left(t\_2 + t\_0 \cdot \cos y\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_3 \cdot \sqrt{2}\right) \cdot \left(t\_1 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + 0.5 \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 3.0 (sqrt 5.0)))
       (t_1 (- (cos x) 1.0))
       (t_2 (- (sqrt 5.0) 1.0))
       (t_3 (- 0.5 (* 0.5 (cos (* 2.0 x))))))
  (if (<= x -175000000.0)
    (*
     0.3333333333333333
     (/
      1.0
      (/
       (- (* (- (* t_2 (cos x)) (- (sqrt 5.0) 3.0)) 0.5) -1.0)
       (- (* (* -0.0625 (* t_1 (sqrt 2.0))) t_3) -2.0))))
    (if (<= x 1.45e-5)
      (*
       (*
        (-
         (*
          (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         -2.0)
        0.3333333333333333)
       (/ 1.0 (- (* 0.5 (+ t_2 (* t_0 (cos y)))) -1.0)))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_3 (sqrt 2.0)) (* t_1 -0.0625)))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_2)) (* 0.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - 1.0;
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_1 * sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_2 + (t_0 * cos(y)))) - -1.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * 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) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = cos(x) - 1.0d0
    t_2 = sqrt(5.0d0) - 1.0d0
    t_3 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    if (x <= (-175000000.0d0)) then
        tmp = 0.3333333333333333d0 * (1.0d0 / (((((t_2 * cos(x)) - (sqrt(5.0d0) - 3.0d0)) * 0.5d0) - (-1.0d0)) / ((((-0.0625d0) * (t_1 * sqrt(2.0d0))) * t_3) - (-2.0d0))))
    else if (x <= 1.45d-5) then
        tmp = (((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) * 0.3333333333333333d0) * (1.0d0 / ((0.5d0 * (t_2 + (t_0 * cos(y)))) - (-1.0d0)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_3 * sqrt(2.0d0)) * (t_1 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + (0.5d0 * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.cos(x) - 1.0;
	double t_2 = Math.sqrt(5.0) - 1.0;
	double t_3 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double tmp;
	if (x <= -175000000.0) {
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_1 * Math.sqrt(2.0))) * t_3) - -2.0)));
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_2 + (t_0 * Math.cos(y)))) - -1.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * Math.sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_2)) + (0.5 * t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.cos(x) - 1.0
	t_2 = math.sqrt(5.0) - 1.0
	t_3 = 0.5 - (0.5 * math.cos((2.0 * x)))
	tmp = 0
	if x <= -175000000.0:
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * math.cos(x)) - (math.sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_1 * math.sqrt(2.0))) * t_3) - -2.0)))
	elif x <= 1.45e-5:
		tmp = ((((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_2 + (t_0 * math.cos(y)))) - -1.0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * math.sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_2)) + (0.5 * t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - 1.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_2 * cos(x)) - Float64(sqrt(5.0) - 3.0)) * 0.5) - -1.0) / Float64(Float64(Float64(-0.0625 * Float64(t_1 * sqrt(2.0))) * t_3) - -2.0))));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) * 0.3333333333333333) * Float64(1.0 / Float64(Float64(0.5 * Float64(t_2 + Float64(t_0 * cos(y)))) - -1.0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_3 * sqrt(2.0)) * Float64(t_1 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + Float64(0.5 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = cos(x) - 1.0;
	t_2 = sqrt(5.0) - 1.0;
	t_3 = 0.5 - (0.5 * cos((2.0 * x)));
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = 0.3333333333333333 * (1.0 / (((((t_2 * cos(x)) - (sqrt(5.0) - 3.0)) * 0.5) - -1.0) / (((-0.0625 * (t_1 * sqrt(2.0))) * t_3) - -2.0)));
	elseif (x <= 1.45e-5)
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_2 + (t_0 * cos(y)))) - -1.0));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_3 * sqrt(2.0)) * (t_1 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_2)) + (0.5 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[(N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[(N[(-0.0625 * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(1.0 / N[(N[(0.5 * N[(t$95$2 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{\left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) \cdot 0.5 - -1}{\left(-0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - -2}}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\left(\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 77.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - 1\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;\frac{\left(\left(0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0 - 2\right) \cdot 0.3333333333333333}{-0.5 \cdot \left(t\_3 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{0.5 \cdot \left(t\_3 + t\_1 \cdot \cos y\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_0 \cdot \sqrt{2}\right) \cdot \left(t\_2 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_3\right) + 0.5 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
       (t_1 (- 3.0 (sqrt 5.0)))
       (t_2 (- (cos x) 1.0))
       (t_3 (- (sqrt 5.0) 1.0)))
  (if (<= x -175000000.0)
    (/
     (*
      (- (* (* 0.0625 (* t_2 (sqrt 2.0))) t_0) 2.0)
      0.3333333333333333)
     (- (* -0.5 (- (* t_3 (cos x)) (- (sqrt 5.0) 3.0))) 1.0))
    (if (<= x 1.45e-5)
      (*
       (*
        (-
         (*
          (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         -2.0)
        0.3333333333333333)
       (/ 1.0 (- (* 0.5 (+ t_3 (* t_1 (cos y)))) -1.0)))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_0 (sqrt 2.0)) (* t_2 -0.0625)))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_3)) (* 0.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - 1.0;
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = ((((0.0625 * (t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_3 + (t_1 * cos(y)))) - -1.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_3)) + (0.5 * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = cos(x) - 1.0d0
    t_3 = sqrt(5.0d0) - 1.0d0
    if (x <= (-175000000.0d0)) then
        tmp = ((((0.0625d0 * (t_2 * sqrt(2.0d0))) * t_0) - 2.0d0) * 0.3333333333333333d0) / (((-0.5d0) * ((t_3 * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0)
    else if (x <= 1.45d-5) then
        tmp = (((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) * 0.3333333333333333d0) * (1.0d0 / ((0.5d0 * (t_3 + (t_1 * cos(y)))) - (-1.0d0)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_0 * sqrt(2.0d0)) * (t_2 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_3)) + (0.5d0 * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.cos(x) - 1.0;
	double t_3 = Math.sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = ((((0.0625 * (t_2 * Math.sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0);
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_3 + (t_1 * Math.cos(y)))) - -1.0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * Math.sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_3)) + (0.5 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 - (0.5 * math.cos((2.0 * x)))
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.cos(x) - 1.0
	t_3 = math.sqrt(5.0) - 1.0
	tmp = 0
	if x <= -175000000.0:
		tmp = ((((0.0625 * (t_2 * math.sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0)
	elif x <= 1.45e-5:
		tmp = ((((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_3 + (t_1 * math.cos(y)))) - -1.0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * math.sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_3)) + (0.5 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - 1.0)
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.0625 * Float64(t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / Float64(Float64(-0.5 * Float64(Float64(t_3 * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) * 0.3333333333333333) * Float64(1.0 / Float64(Float64(0.5 * Float64(t_3 + Float64(t_1 * cos(y)))) - -1.0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_0 * sqrt(2.0)) * Float64(t_2 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_3)) + Float64(0.5 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	t_1 = 3.0 - sqrt(5.0);
	t_2 = cos(x) - 1.0;
	t_3 = sqrt(5.0) - 1.0;
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = ((((0.0625 * (t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
	elseif (x <= 1.45e-5)
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) * (1.0 / ((0.5 * (t_3 + (t_1 * cos(y)))) - -1.0));
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_3)) + (0.5 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(N[(N[(N[(N[(0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(-0.5 * N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(1.0 / N[(N[(0.5 * N[(t$95$3 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot 0.3333333333333333}{\color{blue}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\left(\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 31: 77.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - 1\\ t_3 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;\frac{\left(\left(0.0625 \cdot \left(t\_2 \cdot \sqrt{2}\right)\right) \cdot t\_0 - 2\right) \cdot 0.3333333333333333}{-0.5 \cdot \left(t\_3 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}{\left(0.5 \cdot \left(t\_3 + t\_1 \cdot \cos y\right) - -1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + \left(t\_0 \cdot \sqrt{2}\right) \cdot \left(t\_2 \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot t\_3\right) + 0.5 \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 x)))))
       (t_1 (- 3.0 (sqrt 5.0)))
       (t_2 (- (cos x) 1.0))
       (t_3 (- (sqrt 5.0) 1.0)))
  (if (<= x -175000000.0)
    (/
     (*
      (- (* (* 0.0625 (* t_2 (sqrt 2.0))) t_0) 2.0)
      0.3333333333333333)
     (- (* -0.5 (- (* t_3 (cos x)) (- (sqrt 5.0) 3.0))) 1.0))
    (if (<= x 1.45e-5)
      (/
       (-
        (*
         (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
         (- 0.5 (* 0.5 (cos (* 2.0 y)))))
        -2.0)
       (* (- (* 0.5 (+ t_3 (* t_1 (cos y)))) -1.0) 3.0))
      (*
       0.3333333333333333
       (/
        (+ 2.0 (* (* t_0 (sqrt 2.0)) (* t_2 -0.0625)))
        (+ 1.0 (+ (* 0.5 (* (cos x) t_3)) (* 0.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - 1.0;
	double t_3 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = ((((0.0625 * (t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) / (((0.5 * (t_3 + (t_1 * cos(y)))) - -1.0) * 3.0);
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_3)) + (0.5 * 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 - (0.5d0 * cos((2.0d0 * x)))
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = cos(x) - 1.0d0
    t_3 = sqrt(5.0d0) - 1.0d0
    if (x <= (-175000000.0d0)) then
        tmp = ((((0.0625d0 * (t_2 * sqrt(2.0d0))) * t_0) - 2.0d0) * 0.3333333333333333d0) / (((-0.5d0) * ((t_3 * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0)
    else if (x <= 1.45d-5) then
        tmp = ((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) / (((0.5d0 * (t_3 + (t_1 * cos(y)))) - (-1.0d0)) * 3.0d0)
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((t_0 * sqrt(2.0d0)) * (t_2 * (-0.0625d0)))) / (1.0d0 + ((0.5d0 * (cos(x) * t_3)) + (0.5d0 * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 - (0.5 * Math.cos((2.0 * x)));
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.cos(x) - 1.0;
	double t_3 = Math.sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = ((((0.0625 * (t_2 * Math.sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0);
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) / (((0.5 * (t_3 + (t_1 * Math.cos(y)))) - -1.0) * 3.0);
	} else {
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * Math.sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (Math.cos(x) * t_3)) + (0.5 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 - (0.5 * math.cos((2.0 * x)))
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.cos(x) - 1.0
	t_3 = math.sqrt(5.0) - 1.0
	tmp = 0
	if x <= -175000000.0:
		tmp = ((((0.0625 * (t_2 * math.sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0)
	elif x <= 1.45e-5:
		tmp = (((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) / (((0.5 * (t_3 + (t_1 * math.cos(y)))) - -1.0) * 3.0)
	else:
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * math.sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (math.cos(x) * t_3)) + (0.5 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * x))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - 1.0)
	t_3 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.0625 * Float64(t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / Float64(Float64(-0.5 * Float64(Float64(t_3 * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0));
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) / Float64(Float64(Float64(0.5 * Float64(t_3 + Float64(t_1 * cos(y)))) - -1.0) * 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(Float64(t_0 * sqrt(2.0)) * Float64(t_2 * -0.0625))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_3)) + Float64(0.5 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 - (0.5 * cos((2.0 * x)));
	t_1 = 3.0 - sqrt(5.0);
	t_2 = cos(x) - 1.0;
	t_3 = sqrt(5.0) - 1.0;
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = ((((0.0625 * (t_2 * sqrt(2.0))) * t_0) - 2.0) * 0.3333333333333333) / ((-0.5 * ((t_3 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0);
	elseif (x <= 1.45e-5)
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) / (((0.5 * (t_3 + (t_1 * cos(y)))) - -1.0) * 3.0);
	else
		tmp = 0.3333333333333333 * ((2.0 + ((t_0 * sqrt(2.0)) * (t_2 * -0.0625))) / (1.0 + ((0.5 * (cos(x) * t_3)) + (0.5 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(N[(N[(N[(N[(0.0625 * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(-0.5 * N[(N[(t$95$3 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[(t$95$3 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot 0.3333333333333333}{\color{blue}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}{\left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1\right) \cdot 3}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{1}{3} \cdot \frac{2 + \left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\sin x \cdot \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{2 + \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]

      18. lower-*.f64 59.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   6. Applied rewrites 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot -0.0625\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 32: 77.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\\ t_2 := -0.5 \cdot \left(t\_0 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;\frac{t\_1 \cdot 0.3333333333333333}{t\_2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}{\left(0.5 \cdot \left(t\_0 + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1
        (-
         (*
          (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         2.0))
       (t_2 (- (* -0.5 (- (* t_0 (cos x)) (- (sqrt 5.0) 3.0))) 1.0)))
  (if (<= x -175000000.0)
    (/ (* t_1 0.3333333333333333) t_2)
    (if (<= x 1.45e-5)
      (/
       (-
        (*
         (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
         (- 0.5 (* 0.5 (cos (* 2.0 y)))))
        -2.0)
       (*
        (- (* 0.5 (+ t_0 (* (- 3.0 (sqrt 5.0)) (cos y)))) -1.0)
        3.0))
      (* (/ t_1 t_2) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) / (((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0) * 3.0);
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	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 = ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - 2.0d0
    t_2 = ((-0.5d0) * ((t_0 * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0
    if (x <= (-175000000.0d0)) then
        tmp = (t_1 * 0.3333333333333333d0) / t_2
    else if (x <= 1.45d-5) then
        tmp = ((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) / (((0.5d0 * (t_0 + ((3.0d0 - sqrt(5.0d0)) * cos(y)))) - (-1.0d0)) * 3.0d0)
    else
        tmp = (t_1 / t_2) * 0.3333333333333333d0
    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.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) / (((0.5 * (t_0 + ((3.0 - Math.sqrt(5.0)) * Math.cos(y)))) - -1.0) * 3.0);
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - 2.0
	t_2 = (-0.5 * ((t_0 * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0
	tmp = 0
	if x <= -175000000.0:
		tmp = (t_1 * 0.3333333333333333) / t_2
	elif x <= 1.45e-5:
		tmp = (((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) / (((0.5 * (t_0 + ((3.0 - math.sqrt(5.0)) * math.cos(y)))) - -1.0) * 3.0)
	else:
		tmp = (t_1 / t_2) * 0.3333333333333333
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = 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)
	t_2 = Float64(Float64(-0.5 * Float64(Float64(t_0 * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(Float64(t_1 * 0.3333333333333333) / t_2);
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) / Float64(Float64(Float64(0.5 * Float64(t_0 + Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))) - -1.0) * 3.0));
	else
		tmp = Float64(Float64(t_1 / t_2) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = (t_1 * 0.3333333333333333) / t_2;
	elseif (x <= 1.45e-5)
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) / (((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0) * 3.0);
	else
		tmp = (t_1 / t_2) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(N[(t$95$1 * 0.3333333333333333), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[(t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot 0.3333333333333333}{\color{blue}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2}{\left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1\right) \cdot 3}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \color{blue}{0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 33: 77.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\\ t_2 := -0.5 \cdot \left(t\_0 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;\frac{t\_1 \cdot 0.3333333333333333}{t\_2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333}{0.5 \cdot \left(t\_0 + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1
        (-
         (*
          (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         2.0))
       (t_2 (- (* -0.5 (- (* t_0 (cos x)) (- (sqrt 5.0) 3.0))) 1.0)))
  (if (<= x -175000000.0)
    (/ (* t_1 0.3333333333333333) t_2)
    (if (<= x 1.45e-5)
      (/
       (*
        (-
         (*
          (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 y)))))
         -2.0)
        0.3333333333333333)
       (- (* 0.5 (+ t_0 (* (- 3.0 (sqrt 5.0)) (cos y)))) -1.0))
      (* (/ t_1 t_2) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) / ((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0);
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	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 = ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - 2.0d0
    t_2 = ((-0.5d0) * ((t_0 * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0
    if (x <= (-175000000.0d0)) then
        tmp = (t_1 * 0.3333333333333333d0) / t_2
    else if (x <= 1.45d-5) then
        tmp = (((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) * 0.3333333333333333d0) / ((0.5d0 * (t_0 + ((3.0d0 - sqrt(5.0d0)) * cos(y)))) - (-1.0d0))
    else
        tmp = (t_1 / t_2) * 0.3333333333333333d0
    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.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = ((((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) / ((0.5 * (t_0 + ((3.0 - Math.sqrt(5.0)) * Math.cos(y)))) - -1.0);
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - 2.0
	t_2 = (-0.5 * ((t_0 * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0
	tmp = 0
	if x <= -175000000.0:
		tmp = (t_1 * 0.3333333333333333) / t_2
	elif x <= 1.45e-5:
		tmp = ((((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) * 0.3333333333333333) / ((0.5 * (t_0 + ((3.0 - math.sqrt(5.0)) * math.cos(y)))) - -1.0)
	else:
		tmp = (t_1 / t_2) * 0.3333333333333333
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = 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)
	t_2 = Float64(Float64(-0.5 * Float64(Float64(t_0 * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(Float64(t_1 * 0.3333333333333333) / t_2);
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) * 0.3333333333333333) / Float64(Float64(0.5 * Float64(t_0 + Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))) - -1.0));
	else
		tmp = Float64(Float64(t_1 / t_2) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = (t_1 * 0.3333333333333333) / t_2;
	elseif (x <= 1.45e-5)
		tmp = ((((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * 0.3333333333333333) / ((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0);
	else
		tmp = (t_1 / t_2) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(N[(t$95$1 * 0.3333333333333333), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot 0.3333333333333333}{\color{blue}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot 0.3333333333333333}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \color{blue}{0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 34: 77.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\\ t_2 := -0.5 \cdot \left(t\_0 \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1\\ \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;\frac{t\_1 \cdot 0.3333333333333333}{t\_2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot \frac{0.3333333333333333}{0.5 \cdot \left(t\_0 + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (sqrt 5.0) 1.0))
       (t_1
        (-
         (*
          (* 0.0625 (* (- (cos x) 1.0) (sqrt 2.0)))
          (- 0.5 (* 0.5 (cos (* 2.0 x)))))
         2.0))
       (t_2 (- (* -0.5 (- (* t_0 (cos x)) (- (sqrt 5.0) 3.0))) 1.0)))
  (if (<= x -175000000.0)
    (/ (* t_1 0.3333333333333333) t_2)
    (if (<= x 1.45e-5)
      (*
       (-
        (*
         (* -0.0625 (* (- 1.0 (cos y)) (sqrt 2.0)))
         (- 0.5 (* 0.5 (cos (* 2.0 y)))))
        -2.0)
       (/
        0.3333333333333333
        (- (* 0.5 (+ t_0 (* (- 3.0 (sqrt 5.0)) (cos y)))) -1.0)))
      (* (/ t_1 t_2) 0.3333333333333333)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * (0.3333333333333333 / ((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0));
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	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 = ((0.0625d0 * ((cos(x) - 1.0d0) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * x))))) - 2.0d0
    t_2 = ((-0.5d0) * ((t_0 * cos(x)) - (sqrt(5.0d0) - 3.0d0))) - 1.0d0
    if (x <= (-175000000.0d0)) then
        tmp = (t_1 * 0.3333333333333333d0) / t_2
    else if (x <= 1.45d-5) then
        tmp = ((((-0.0625d0) * ((1.0d0 - cos(y)) * sqrt(2.0d0))) * (0.5d0 - (0.5d0 * cos((2.0d0 * y))))) - (-2.0d0)) * (0.3333333333333333d0 / ((0.5d0 * (t_0 + ((3.0d0 - sqrt(5.0d0)) * cos(y)))) - (-1.0d0)))
    else
        tmp = (t_1 / t_2) * 0.3333333333333333d0
    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.cos(x) - 1.0) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * x))))) - 2.0;
	double t_2 = (-0.5 * ((t_0 * Math.cos(x)) - (Math.sqrt(5.0) - 3.0))) - 1.0;
	double tmp;
	if (x <= -175000000.0) {
		tmp = (t_1 * 0.3333333333333333) / t_2;
	} else if (x <= 1.45e-5) {
		tmp = (((-0.0625 * ((1.0 - Math.cos(y)) * Math.sqrt(2.0))) * (0.5 - (0.5 * Math.cos((2.0 * y))))) - -2.0) * (0.3333333333333333 / ((0.5 * (t_0 + ((3.0 - Math.sqrt(5.0)) * Math.cos(y)))) - -1.0));
	} else {
		tmp = (t_1 / t_2) * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) - 1.0
	t_1 = ((0.0625 * ((math.cos(x) - 1.0) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * x))))) - 2.0
	t_2 = (-0.5 * ((t_0 * math.cos(x)) - (math.sqrt(5.0) - 3.0))) - 1.0
	tmp = 0
	if x <= -175000000.0:
		tmp = (t_1 * 0.3333333333333333) / t_2
	elif x <= 1.45e-5:
		tmp = (((-0.0625 * ((1.0 - math.cos(y)) * math.sqrt(2.0))) * (0.5 - (0.5 * math.cos((2.0 * y))))) - -2.0) * (0.3333333333333333 / ((0.5 * (t_0 + ((3.0 - math.sqrt(5.0)) * math.cos(y)))) - -1.0))
	else:
		tmp = (t_1 / t_2) * 0.3333333333333333
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = 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)
	t_2 = Float64(Float64(-0.5 * Float64(Float64(t_0 * cos(x)) - Float64(sqrt(5.0) - 3.0))) - 1.0)
	tmp = 0.0
	if (x <= -175000000.0)
		tmp = Float64(Float64(t_1 * 0.3333333333333333) / t_2);
	elseif (x <= 1.45e-5)
		tmp = Float64(Float64(Float64(Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * y))))) - -2.0) * Float64(0.3333333333333333 / Float64(Float64(0.5 * Float64(t_0 + Float64(Float64(3.0 - sqrt(5.0)) * cos(y)))) - -1.0)));
	else
		tmp = Float64(Float64(t_1 / t_2) * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) - 1.0;
	t_1 = ((0.0625 * ((cos(x) - 1.0) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * x))))) - 2.0;
	t_2 = (-0.5 * ((t_0 * cos(x)) - (sqrt(5.0) - 3.0))) - 1.0;
	tmp = 0.0;
	if (x <= -175000000.0)
		tmp = (t_1 * 0.3333333333333333) / t_2;
	elseif (x <= 1.45e-5)
		tmp = (((-0.0625 * ((1.0 - cos(y)) * sqrt(2.0))) * (0.5 - (0.5 * cos((2.0 * y))))) - -2.0) * (0.3333333333333333 / ((0.5 * (t_0 + ((3.0 - sqrt(5.0)) * cos(y)))) - -1.0));
	else
		tmp = (t_1 / t_2) * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -175000000.0], N[(N[(t$95$1 * 0.3333333333333333), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 1.45e-5], N[(N[(N[(N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(0.5 * N[(t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e8

   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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2\right) \cdot 0.3333333333333333}{\color{blue}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1}} \]

   if -1.75e8 < x < 1.45e-5

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - \color{blue}{1}\right)\right)\right)} \]

      10. lower-sqrt.f64 60.1%

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. Applied rewrites 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{\color{blue}{2}} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(\color{blue}{1} - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right)\right)}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{2 + \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   9. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\left(\left(-0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot y\right)\right) - -2\right) \cdot \frac{0.3333333333333333}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right) \cdot \cos y\right) - -1}} \]

   if 1.45e-5 < 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 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   6. Applied rewrites 59.4%

      \[\leadsto \frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \color{blue}{0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 35: 59.4% accurate, 2.3× speedup

Math

\[\frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot 0.3333333333333333 \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (/
  (-
   (*
    (* 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))

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[(0.0625 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(-0.5 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. 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 59.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

6. Applied rewrites 59.4%

   \[\leadsto \frac{\left(0.0625 \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) - 2}{-0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x - \left(\sqrt{5} - 3\right)\right) - 1} \cdot \color{blue}{0.3333333333333333} \]
7. Add Preprocessing

Alternative 36: 43.0% accurate, 5.7× speedup

Math

\[0.3333333333333333 \cdot \frac{2}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 0.3333333333333333
 (/
  2.0
  (+
   1.0
   (+
    (* 0.5 (* (cos x) (- (sqrt 5.0) 1.0)))
    (* 0.5 (- 3.0 (sqrt 5.0))))))))

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[(0.3333333333333333 * N[(2.0 / N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. 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 59.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 43.0%

   \[\leadsto 0.3333333333333333 \cdot \frac{2}{\color{blue}{1} + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
8. Add Preprocessing

Alternative 37: 40.5% accurate, 940.0× speedup

Math

\[0.3333333333333333 \]

FPCore

(FPCore (x y)
  :precision binary64
  0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
3. Step-by-step derivation

   1. 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 59.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

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 40.5%

      \[\leadsto \frac{0.6666666666666666}{1 + \left(0.5 \cdot \left(3 - \sqrt{5}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)} \]

7. Applied rewrites 40.5%

   \[\leadsto \frac{0.6666666666666666}{\color{blue}{1 + \left(0.5 \cdot \left(3 - \sqrt{5}\right) + 0.5 \cdot \left(\sqrt{5} - 1\right)\right)}} \]

8. Evaluated real constant 40.5%

   \[\leadsto 0.3333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))