Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 68.5% → 99.8%

Time: 4.0s

Alternatives: 24

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (* (/ (/ x (+ y x)) (- (+ y x) -1.0)) (/ y (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	return ((x / (y + x)) / ((y + x) - -1.0)) * (y / (y + x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = ((x / (y + x)) / ((y + x) - (-1.0d0))) * (y / (y + x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return ((x / (y + x)) / ((y + x) - -1.0)) * (y / (y + x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((x / (y + x)) / ((y + x) - -1.0)) * (y / (y + x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) - -1.0)) * Float64(y / Float64(y + x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((x / (y + x)) / ((y + x) - -1.0)) * (y / (y + x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   14. lower-*.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

   16. add-flip N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   17. lower--.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   19. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   20. lower-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   21. metadata-eval 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

   22. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   23. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   24. lower-+.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

3. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

   8. lower-/.f64 99.8

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
6. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{y + x}}{\frac{x - \left(-1 - y\right)}{y} \cdot \left(y + x\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ y x)) (* (/ (- x (- -1.0 y)) y) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	return (x / (y + x)) / (((x - (-1.0 - y)) / y) * (y + x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = (x / (y + x)) / (((x - ((-1.0d0) - y)) / y) * (y + x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (x / (y + x)) / (((x - (-1.0 - y)) / y) * (y + x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (y + x)) / (((x - (-1.0 - y)) / y) * (y + x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(y + x)) / Float64(Float64(Float64(x - Float64(-1.0 - y)) / y) * Float64(y + x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (y + x)) / (((x - (-1.0 - y)) / y) * (y + x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   14. lower-*.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

   16. add-flip N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   17. lower--.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   19. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   20. lower-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   21. metadata-eval 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

   22. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   23. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   24. lower-+.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

3. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

   8. lower-/.f64 99.8

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

   6. frac-times N/A

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

   10. +-commutative N/A

       \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

   14. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

   16. frac-times N/A

       \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

   17. div-flip N/A

       \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

   18. lift-/.f64 N/A

       \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{\left(y + x\right) - -1}}{y} \cdot \left(y + x\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{\left(y + x\right)} - -1}{y} \cdot \left(y + x\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{\left(x + y\right)} - -1}{y} \cdot \left(y + x\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\left(x + y\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}{y} \cdot \left(y + x\right)} \]

   5. add-flip N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{\left(x + y\right) + 1}}{y} \cdot \left(y + x\right)} \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(y + x\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + \color{blue}{\left(1 + y\right)}}{y} \cdot \left(y + x\right)} \]

   8. lift-+.f64 N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{x + \color{blue}{\left(1 + y\right)}}{y} \cdot \left(y + x\right)} \]

   9. add-flip N/A

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(\left(1 + y\right)\right)\right)}}{y} \cdot \left(y + x\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{x - \left(\mathsf{neg}\left(\left(1 + y\right)\right)\right)}}{y} \cdot \left(y + x\right)} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{x - \left(\mathsf{neg}\left(\color{blue}{\left(1 + y\right)}\right)\right)}{y} \cdot \left(y + x\right)} \]

   12. distribute-neg-in N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{x - \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)}}{y} \cdot \left(y + x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{x - \left(\color{blue}{-1} + \left(\mathsf{neg}\left(y\right)\right)\right)}{y} \cdot \left(y + x\right)} \]

   14. sub-flip N/A

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{x - \color{blue}{\left(-1 - y\right)}}{y} \cdot \left(y + x\right)} \]

   15. lower--.f64 99.2

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{x - \color{blue}{\left(-1 - y\right)}}{y} \cdot \left(y + x\right)} \]

9. Applied rewrites 99.2%

   \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{x - \left(-1 - y\right)}}{y} \cdot \left(y + x\right)} \]
10. Add Preprocessing

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq -1.95 \cdot 10^{+36}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+153}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y -1.95e+36)
     (/ (* t_0 (/ y (- x -1.0))) (+ y x))
     (if (<= y 1.6e+153)
       (* t_0 (/ y (* (- (+ y x) -1.0) (+ y x))))
       (/ t_0 (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= -1.95e+36) {
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	} else if (y <= 1.6e+153) {
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = t_0 / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (y + x)
    if (y <= (-1.95d+36)) then
        tmp = (t_0 * (y / (x - (-1.0d0)))) / (y + x)
    else if (y <= 1.6d+153) then
        tmp = t_0 * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = t_0 / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= -1.95e+36) {
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	} else if (y <= 1.6e+153) {
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = t_0 / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= -1.95e+36:
		tmp = (t_0 * (y / (x - -1.0))) / (y + x)
	elif y <= 1.6e+153:
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = t_0 / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= -1.95e+36)
		tmp = Float64(Float64(t_0 * Float64(y / Float64(x - -1.0))) / Float64(y + x));
	elseif (y <= 1.6e+153)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(t_0 / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= -1.95e+36)
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	elseif (y <= 1.6e+153)
		tmp = t_0 * (y / (((y + x) - -1.0) * (y + x)));
	else
		tmp = t_0 / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.95e+36], N[(N[(t$95$0 * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+153], N[(t$95$0 * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9500000000000001e36

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      15. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{1 + x}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}} \]

   if -1.9500000000000001e36 < y < 1.6000000000000001e153

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   if 1.6000000000000001e153 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{y + x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 7.8e-23)
     (/ (* t_0 (/ y (- x -1.0))) (+ y x))
     (if (<= y 1.6e+153)
       (* (/ (/ y (* (- (+ y x) -1.0) (+ y x))) (+ y x)) x)
       (/ t_0 (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 7.8e-23) {
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	} else if (y <= 1.6e+153) {
		tmp = ((y / (((y + x) - -1.0) * (y + x))) / (y + x)) * x;
	} else {
		tmp = t_0 / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (y + x)
    if (y <= 7.8d-23) then
        tmp = (t_0 * (y / (x - (-1.0d0)))) / (y + x)
    else if (y <= 1.6d+153) then
        tmp = ((y / (((y + x) - (-1.0d0)) * (y + x))) / (y + x)) * x
    else
        tmp = t_0 / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 7.8e-23) {
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	} else if (y <= 1.6e+153) {
		tmp = ((y / (((y + x) - -1.0) * (y + x))) / (y + x)) * x;
	} else {
		tmp = t_0 / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 7.8e-23:
		tmp = (t_0 * (y / (x - -1.0))) / (y + x)
	elif y <= 1.6e+153:
		tmp = ((y / (((y + x) - -1.0) * (y + x))) / (y + x)) * x
	else:
		tmp = t_0 / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 7.8e-23)
		tmp = Float64(Float64(t_0 * Float64(y / Float64(x - -1.0))) / Float64(y + x));
	elseif (y <= 1.6e+153)
		tmp = Float64(Float64(Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))) / Float64(y + x)) * x);
	else
		tmp = Float64(t_0 / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 7.8e-23)
		tmp = (t_0 * (y / (x - -1.0))) / (y + x);
	elseif (y <= 1.6e+153)
		tmp = ((y / (((y + x) - -1.0) * (y + x))) / (y + x)) * x;
	else
		tmp = t_0 / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.8e-23], N[(N[(t$95$0 * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+153], N[(N[(N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(t$95$0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.8e-23

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      15. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{1 + x}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}} \]

   if 7.8e-23 < y < 1.6000000000000001e153

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   3. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \cdot x \]

      4. associate-*r* N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)}} \cdot x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)} \cdot \left(y + x\right)} \cdot x \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{y + x}} \cdot x \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}}{y + x} \cdot x \]

      8. lower-/.f64 90.6

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{y + x}} \cdot x \]

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}}{y + x}} \cdot x \]

   if 1.6000000000000001e153 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) - -1\\ \mathbf{if}\;y \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{t\_0}{y} \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x) -1.0)))
   (if (<= y 1.02e-29)
     (/ (* (/ x (+ y x)) (/ y (- x -1.0))) (+ y x))
     (if (<= y 3.1e+86)
       (* (/ y (* t_0 (* (+ y x) (+ y x)))) x)
       (/ (/ x y) (* (/ t_0 y) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.02e-29) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else if (y <= 3.1e+86) {
		tmp = (y / (t_0 * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (x / y) / ((t_0 / y) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = (y + x) - (-1.0d0)
    if (y <= 1.02d-29) then
        tmp = ((x / (y + x)) * (y / (x - (-1.0d0)))) / (y + x)
    else if (y <= 3.1d+86) then
        tmp = (y / (t_0 * ((y + x) * (y + x)))) * x
    else
        tmp = (x / y) / ((t_0 / y) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.02e-29) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else if (y <= 3.1e+86) {
		tmp = (y / (t_0 * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (x / y) / ((t_0 / y) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) - -1.0
	tmp = 0
	if y <= 1.02e-29:
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x)
	elif y <= 3.1e+86:
		tmp = (y / (t_0 * ((y + x) * (y + x)))) * x
	else:
		tmp = (x / y) / ((t_0 / y) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) - -1.0)
	tmp = 0.0
	if (y <= 1.02e-29)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(x - -1.0))) / Float64(y + x));
	elseif (y <= 3.1e+86)
		tmp = Float64(Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))) * x);
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(t_0 / y) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) - -1.0;
	tmp = 0.0;
	if (y <= 1.02e-29)
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	elseif (y <= 3.1e+86)
		tmp = (y / (t_0 * ((y + x) * (y + x)))) * x;
	else
		tmp = (x / y) / ((t_0 / y) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, 1.02e-29], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+86], N[(N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(t$95$0 / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999994e-29

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      15. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{1 + x}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}} \]

   if 1.01999999999999994e-29 < y < 3.1000000000000002e86

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   3. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   if 3.1000000000000002e86 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 61.8

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]

   10. Applied rewrites 61.8%

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

Alternative 6: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x - \left(-1 - y\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e-29)
   (/ (* (/ x (+ y x)) (/ y (- x -1.0))) (+ y x))
   (if (<= y 3.1e+86)
     (* (/ y (* (* (+ y x) (+ y x)) (- x (- -1.0 y)))) x)
     (/ (/ x y) (* (/ (- (+ y x) -1.0) y) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-29) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else if (y <= 3.1e+86) {
		tmp = (y / (((y + x) * (y + x)) * (x - (-1.0 - y)))) * x;
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 1.02d-29) then
        tmp = ((x / (y + x)) * (y / (x - (-1.0d0)))) / (y + x)
    else if (y <= 3.1d+86) then
        tmp = (y / (((y + x) * (y + x)) * (x - ((-1.0d0) - y)))) * x
    else
        tmp = (x / y) / ((((y + x) - (-1.0d0)) / y) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e-29) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else if (y <= 3.1e+86) {
		tmp = (y / (((y + x) * (y + x)) * (x - (-1.0 - y)))) * x;
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e-29:
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x)
	elif y <= 3.1e+86:
		tmp = (y / (((y + x) * (y + x)) * (x - (-1.0 - y)))) * x
	else:
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e-29)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(x - -1.0))) / Float64(y + x));
	elseif (y <= 3.1e+86)
		tmp = Float64(Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x - Float64(-1.0 - y)))) * x);
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(Float64(y + x) - -1.0) / y) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e-29)
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	elseif (y <= 3.1e+86)
		tmp = (y / (((y + x) * (y + x)) * (x - (-1.0 - y)))) * x;
	else
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.02e-29], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+86], N[(N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999994e-29

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      15. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{1 + x}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}} \]

   if 1.01999999999999994e-29 < y < 3.1000000000000002e86

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   9. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x - \left(-1 - y\right)\right)} \cdot x} \]

   if 3.1000000000000002e86 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 61.8

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]

   10. Applied rewrites 61.8%

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

Alternative 7: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-10)
   (/ (* (/ x (+ y x)) (/ y (- x -1.0))) (+ y x))
   (/ (/ x y) (* (/ (- (+ y x) -1.0) y) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 5.5d-10) then
        tmp = ((x / (y + x)) * (y / (x - (-1.0d0)))) / (y + x)
    else
        tmp = (x / y) / ((((y + x) - (-1.0d0)) / y) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-10:
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x)
	else:
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-10)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(x - -1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(Float64(y + x) - -1.0) / y) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-10)
		tmp = ((x / (y + x)) * (y / (x - -1.0))) / (y + x);
	else
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-10], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999996e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      15. frac-times N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{1 + x}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{x - -1}}{y + x}} \]

   if 5.4999999999999996e-10 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 61.8

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]

   10. Applied rewrites 61.8%

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

Alternative 8: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-10)
   (/ y (* (* (+ y x) (/ (+ y x) x)) (- x -1.0)))
   (if (<= y 3.2e+152)
     (/ (* (/ x (* (+ y x) (+ y x))) y) (+ 1.0 y))
     (/ (/ x (+ y x)) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else if (y <= 3.2e+152) {
		tmp = ((x / ((y + x) * (y + x))) * y) / (1.0 + y);
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 5.5d-10) then
        tmp = y / (((y + x) * ((y + x) / x)) * (x - (-1.0d0)))
    else if (y <= 3.2d+152) then
        tmp = ((x / ((y + x) * (y + x))) * y) / (1.0d0 + y)
    else
        tmp = (x / (y + x)) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else if (y <= 3.2e+152) {
		tmp = ((x / ((y + x) * (y + x))) * y) / (1.0 + y);
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-10:
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0))
	elif y <= 3.2e+152:
		tmp = ((x / ((y + x) * (y + x))) * y) / (1.0 + y)
	else:
		tmp = (x / (y + x)) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-10)
		tmp = Float64(y / Float64(Float64(Float64(y + x) * Float64(Float64(y + x) / x)) * Float64(x - -1.0)));
	elseif (y <= 3.2e+152)
		tmp = Float64(Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * y) / Float64(1.0 + y));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-10)
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	elseif (y <= 3.2e+152)
		tmp = ((x / ((y + x) * (y + x))) * y) / (1.0 + y);
	else
		tmp = (x / (y + x)) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-10], N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4999999999999996e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{1 + x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{y}{1 + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x}}} \cdot \frac{y}{1 + x} \]

      12. frac-times N/A

          \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{y}}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

   6. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}} \]

   if 5.4999999999999996e-10 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 74.9

         \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{1 + \color{blue}{y}} \]

   6. Applied rewrites 74.9%

      \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{1 + y}} \]

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-10)
   (/ y (* (* (+ y x) (/ (+ y x) x)) (- x -1.0)))
   (if (<= y 3.2e+152)
     (* (/ x y) (/ y (* (- (+ y x) -1.0) (+ y x))))
     (/ (/ x (+ y x)) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 5.5d-10) then
        tmp = y / (((y + x) * ((y + x) / x)) * (x - (-1.0d0)))
    else if (y <= 3.2d+152) then
        tmp = (x / y) * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (x / (y + x)) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-10:
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0))
	elif y <= 3.2e+152:
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (x / (y + x)) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-10)
		tmp = Float64(y / Float64(Float64(Float64(y + x) * Float64(Float64(y + x) / x)) * Float64(x - -1.0)));
	elseif (y <= 3.2e+152)
		tmp = Float64(Float64(x / y) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-10)
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	elseif (y <= 3.2e+152)
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (x / (y + x)) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-10], N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(N[(x / y), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4999999999999996e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{1 + x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{y}{1 + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x}}} \cdot \frac{y}{1 + x} \]

      12. frac-times N/A

          \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{y}}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

   6. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}} \]

   if 5.4999999999999996e-10 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 58.7

         \[\leadsto \frac{x}{\color{blue}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   6. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-10)
   (/ y (* (* (+ y x) (/ (+ y x) x)) (- x -1.0)))
   (/ (/ x y) (* (/ (- (+ y x) -1.0) y) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 5.5d-10) then
        tmp = y / (((y + x) * ((y + x) / x)) * (x - (-1.0d0)))
    else
        tmp = (x / y) / ((((y + x) - (-1.0d0)) / y) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	} else {
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-10:
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0))
	else:
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-10)
		tmp = Float64(y / Float64(Float64(Float64(y + x) * Float64(Float64(y + x) / x)) * Float64(x - -1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(Float64(y + x) - -1.0) / y) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-10)
		tmp = y / (((y + x) * ((y + x) / x)) * (x - -1.0));
	else
		tmp = (x / y) / ((((y + x) - -1.0) / y) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-10], N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999996e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{1 + x}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      6. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{1 + x} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{y}{1 + x} \]

      9. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{1 + x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x}}} \cdot \frac{y}{1 + x} \]

      12. frac-times N/A

          \[\leadsto \color{blue}{\frac{1 \cdot y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{y}}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\frac{\left(y + x\right) \cdot \left(y + x\right)}{x} \cdot \left(1 + x\right)}} \]

   6. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) \cdot \frac{y + x}{x}\right) \cdot \left(x - -1\right)}} \]

   if 5.4999999999999996e-10 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 61.8

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)} \]

   10. Applied rewrites 61.8%

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

Alternative 11: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(x - -1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-10)
   (* (/ y (+ y x)) (/ x (* (- x -1.0) (+ y x))))
   (if (<= y 3.2e+152)
     (* (/ x y) (/ y (* (- (+ y x) -1.0) (+ y x))))
     (/ (/ x (+ y x)) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = (y / (y + x)) * (x / ((x - -1.0) * (y + x)));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 5.5d-10) then
        tmp = (y / (y + x)) * (x / ((x - (-1.0d0)) * (y + x)))
    else if (y <= 3.2d+152) then
        tmp = (x / y) * (y / (((y + x) - (-1.0d0)) * (y + x)))
    else
        tmp = (x / (y + x)) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-10) {
		tmp = (y / (y + x)) * (x / ((x - -1.0) * (y + x)));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e-10:
		tmp = (y / (y + x)) * (x / ((x - -1.0) * (y + x)))
	elif y <= 3.2e+152:
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)))
	else:
		tmp = (x / (y + x)) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-10)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(x - -1.0) * Float64(y + x))));
	elseif (y <= 3.2e+152)
		tmp = Float64(Float64(x / y) * Float64(y / Float64(Float64(Float64(y + x) - -1.0) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-10)
		tmp = (y / (y + x)) * (x / ((x - -1.0) * (y + x)));
	elseif (y <= 3.2e+152)
		tmp = (x / y) * (y / (((y + x) - -1.0) * (y + x)));
	else
		tmp = (x / (y + x)) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e-10], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(N[(x / y), $MachinePrecision] * N[(y / N[(N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4999999999999996e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + x\right)\right)} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(1 + x\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(1 + x\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(1 + x\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(y + x\right)}} \]

      18. lower-*.f64 76.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(y + x\right)}} \]

   6. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x - -1\right) \cdot \left(y + x\right)}} \]

   if 5.4999999999999996e-10 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 58.7

         \[\leadsto \frac{x}{\color{blue}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   6. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) - -1} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot y}{\left(y - -1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -7.5e-9)
     (* (/ t_0 (- (+ y x) -1.0)) (/ y x))
     (/ (* t_0 y) (* (- y -1.0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -7.5e-9) {
		tmp = (t_0 / ((y + x) - -1.0)) * (y / x);
	} else {
		tmp = (t_0 * y) / ((y - -1.0) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (y + x)
    if (x <= (-7.5d-9)) then
        tmp = (t_0 / ((y + x) - (-1.0d0))) * (y / x)
    else
        tmp = (t_0 * y) / ((y - (-1.0d0)) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -7.5e-9) {
		tmp = (t_0 / ((y + x) - -1.0)) * (y / x);
	} else {
		tmp = (t_0 * y) / ((y - -1.0) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -7.5e-9:
		tmp = (t_0 / ((y + x) - -1.0)) * (y / x)
	else:
		tmp = (t_0 * y) / ((y - -1.0) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -7.5e-9)
		tmp = Float64(Float64(t_0 / Float64(Float64(y + x) - -1.0)) * Float64(y / x));
	else
		tmp = Float64(Float64(t_0 * y) / Float64(Float64(y - -1.0) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -7.5e-9)
		tmp = (t_0 / ((y + x) - -1.0)) * (y / x);
	else
		tmp = (t_0 * y) / ((y - -1.0) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e-9], N[(N[(t$95$0 / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * y), $MachinePrecision] / N[(N[(y - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999933e-9

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. lower-/.f64 61.6

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{\color{blue}{x}} \]

   8. Applied rewrites 61.6%

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{x}} \]

   if -7.49999999999999933e-9 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 58.2

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(1 + y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(1 + y\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{y + x}}{\left(x + y\right) \cdot \left(1 + y\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{y + x}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      10. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot y}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(x + y\right) \cdot \left(1 + y\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot y}}{\left(x + y\right) \cdot \left(1 + y\right)} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(x + y\right)} \cdot \left(1 + y\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + y\right)} \]

      16. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(1 + y\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(1 + y\right) \cdot \left(y + x\right)}} \]

   6. Applied rewrites 75.6%

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

Alternative 13: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) - -1\\ \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{y}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (+ y x) -1.0)))
   (if (<= y 1.6e-162)
     (/ (/ y x) t_0)
     (if (<= y 4.6e-10)
       (* y (/ x (* (- x -1.0) (* (+ y x) (+ y x)))))
       (if (<= y 3.2e+152)
         (* (/ x y) (/ y (* t_0 (+ y x))))
         (/ (/ x (+ y x)) (+ 1.0 y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.6e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 4.6e-10) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (t_0 * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = (y + x) - (-1.0d0)
    if (y <= 1.6d-162) then
        tmp = (y / x) / t_0
    else if (y <= 4.6d-10) then
        tmp = y * (x / ((x - (-1.0d0)) * ((y + x) * (y + x))))
    else if (y <= 3.2d+152) then
        tmp = (x / y) * (y / (t_0 * (y + x)))
    else
        tmp = (x / (y + x)) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.6e-162) {
		tmp = (y / x) / t_0;
	} else if (y <= 4.6e-10) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else if (y <= 3.2e+152) {
		tmp = (x / y) * (y / (t_0 * (y + x)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) - -1.0
	tmp = 0
	if y <= 1.6e-162:
		tmp = (y / x) / t_0
	elif y <= 4.6e-10:
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))))
	elif y <= 3.2e+152:
		tmp = (x / y) * (y / (t_0 * (y + x)))
	else:
		tmp = (x / (y + x)) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) - -1.0)
	tmp = 0.0
	if (y <= 1.6e-162)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 4.6e-10)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 3.2e+152)
		tmp = Float64(Float64(x / y) * Float64(y / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) - -1.0;
	tmp = 0.0;
	if (y <= 1.6e-162)
		tmp = (y / x) / t_0;
	elseif (y <= 4.6e-10)
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	elseif (y <= 3.2e+152)
		tmp = (x / y) * (y / (t_0 * (y + x)));
	else
		tmp = (x / (y + x)) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, 1.6e-162], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 4.6e-10], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(N[(x / y), $MachinePrecision] * N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 1.59999999999999988e-162 < y < 4.60000000000000014e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      6. lower-/.f64 75.6

         \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      9. lower-*.f64 75.6

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(1 + \color{blue}{x}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto y \cdot \frac{x}{\left(x + \color{blue}{1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      12. add-flip-rev N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      14. lower--.f64 75.6

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right) \cdot \left(x + y\right)\right)} \]

      16. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(+-commutative, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      17. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      18. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right)\right)} \]

      19. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(+-commutative, \left(y + x\right)\right)\right)} \]

      20. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right)\right)} \]

   6. Applied rewrites 75.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   if 4.60000000000000014e-10 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 58.7

         \[\leadsto \frac{x}{\color{blue}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   6. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \]

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \left(y + x\right)\\ t_1 := \left(y + x\right) - -1\\ \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_1}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot t\_0}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{y}{\left(y - -1\right) \cdot t\_0} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_1} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (+ y x))) (t_1 (- (+ y x) -1.0)))
   (if (<= y 1.6e-162)
     (/ (/ y x) t_1)
     (if (<= y 4.4e-10)
       (* y (/ x (* (- x -1.0) t_0)))
       (if (<= y 1.5e+118)
         (* (/ y (* (- y -1.0) t_0)) x)
         (* (/ (/ x (+ y x)) t_1) 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double t_1 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.6e-162) {
		tmp = (y / x) / t_1;
	} else if (y <= 4.4e-10) {
		tmp = y * (x / ((x - -1.0) * t_0));
	} else if (y <= 1.5e+118) {
		tmp = (y / ((y - -1.0) * t_0)) * x;
	} else {
		tmp = ((x / (y + x)) / t_1) * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = (y + x) * (y + x)
    t_1 = (y + x) - (-1.0d0)
    if (y <= 1.6d-162) then
        tmp = (y / x) / t_1
    else if (y <= 4.4d-10) then
        tmp = y * (x / ((x - (-1.0d0)) * t_0))
    else if (y <= 1.5d+118) then
        tmp = (y / ((y - (-1.0d0)) * t_0)) * x
    else
        tmp = ((x / (y + x)) / t_1) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double t_1 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.6e-162) {
		tmp = (y / x) / t_1;
	} else if (y <= 4.4e-10) {
		tmp = y * (x / ((x - -1.0) * t_0));
	} else if (y <= 1.5e+118) {
		tmp = (y / ((y - -1.0) * t_0)) * x;
	} else {
		tmp = ((x / (y + x)) / t_1) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) * (y + x)
	t_1 = (y + x) - -1.0
	tmp = 0
	if y <= 1.6e-162:
		tmp = (y / x) / t_1
	elif y <= 4.4e-10:
		tmp = y * (x / ((x - -1.0) * t_0))
	elif y <= 1.5e+118:
		tmp = (y / ((y - -1.0) * t_0)) * x
	else:
		tmp = ((x / (y + x)) / t_1) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(y + x))
	t_1 = Float64(Float64(y + x) - -1.0)
	tmp = 0.0
	if (y <= 1.6e-162)
		tmp = Float64(Float64(y / x) / t_1);
	elseif (y <= 4.4e-10)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * t_0)));
	elseif (y <= 1.5e+118)
		tmp = Float64(Float64(y / Float64(Float64(y - -1.0) * t_0)) * x);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_1) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) * (y + x);
	t_1 = (y + x) - -1.0;
	tmp = 0.0;
	if (y <= 1.6e-162)
		tmp = (y / x) / t_1;
	elseif (y <= 4.4e-10)
		tmp = y * (x / ((x - -1.0) * t_0));
	elseif (y <= 1.5e+118)
		tmp = (y / ((y - -1.0) * t_0)) * x;
	else
		tmp = ((x / (y + x)) / t_1) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, 1.6e-162], N[(N[(y / x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 4.4e-10], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+118], N[(N[(y / N[(N[(y - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 1.59999999999999988e-162 < y < 4.3999999999999998e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      6. lower-/.f64 75.6

         \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      9. lower-*.f64 75.6

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(1 + x\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(1 + \color{blue}{x}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto y \cdot \frac{x}{\left(x + \color{blue}{1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      12. add-flip-rev N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      14. lower--.f64 75.6

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      15. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right) \cdot \left(x + y\right)\right)} \]

      16. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(+-commutative, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      17. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      18. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right)\right)} \]

      19. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(+-commutative, \left(y + x\right)\right)\right)} \]

      20. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right)\right)} \]

   6. Applied rewrites 75.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   if 4.3999999999999998e-10 < y < 1.5e118

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 58.2

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \cdot x} \]

   6. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{y}{\left(y - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   if 1.5e118 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

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

Alternative 15: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \left(y + x\right)\\ t_1 := \left(y + x\right) - -1\\ \mathbf{if}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_1}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot t\_0} \cdot x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{y}{\left(y - -1\right) \cdot t\_0} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_1} \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (+ y x))) (t_1 (- (+ y x) -1.0)))
   (if (<= y 1.95e-162)
     (/ (/ y x) t_1)
     (if (<= y 4.4e-10)
       (* (/ y (* (- x -1.0) t_0)) x)
       (if (<= y 1.5e+118)
         (* (/ y (* (- y -1.0) t_0)) x)
         (* (/ (/ x (+ y x)) t_1) 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double t_1 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.95e-162) {
		tmp = (y / x) / t_1;
	} else if (y <= 4.4e-10) {
		tmp = (y / ((x - -1.0) * t_0)) * x;
	} else if (y <= 1.5e+118) {
		tmp = (y / ((y - -1.0) * t_0)) * x;
	} else {
		tmp = ((x / (y + x)) / t_1) * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = (y + x) * (y + x)
    t_1 = (y + x) - (-1.0d0)
    if (y <= 1.95d-162) then
        tmp = (y / x) / t_1
    else if (y <= 4.4d-10) then
        tmp = (y / ((x - (-1.0d0)) * t_0)) * x
    else if (y <= 1.5d+118) then
        tmp = (y / ((y - (-1.0d0)) * t_0)) * x
    else
        tmp = ((x / (y + x)) / t_1) * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double t_1 = (y + x) - -1.0;
	double tmp;
	if (y <= 1.95e-162) {
		tmp = (y / x) / t_1;
	} else if (y <= 4.4e-10) {
		tmp = (y / ((x - -1.0) * t_0)) * x;
	} else if (y <= 1.5e+118) {
		tmp = (y / ((y - -1.0) * t_0)) * x;
	} else {
		tmp = ((x / (y + x)) / t_1) * 1.0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) * (y + x)
	t_1 = (y + x) - -1.0
	tmp = 0
	if y <= 1.95e-162:
		tmp = (y / x) / t_1
	elif y <= 4.4e-10:
		tmp = (y / ((x - -1.0) * t_0)) * x
	elif y <= 1.5e+118:
		tmp = (y / ((y - -1.0) * t_0)) * x
	else:
		tmp = ((x / (y + x)) / t_1) * 1.0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(y + x))
	t_1 = Float64(Float64(y + x) - -1.0)
	tmp = 0.0
	if (y <= 1.95e-162)
		tmp = Float64(Float64(y / x) / t_1);
	elseif (y <= 4.4e-10)
		tmp = Float64(Float64(y / Float64(Float64(x - -1.0) * t_0)) * x);
	elseif (y <= 1.5e+118)
		tmp = Float64(Float64(y / Float64(Float64(y - -1.0) * t_0)) * x);
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_1) * 1.0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) * (y + x);
	t_1 = (y + x) - -1.0;
	tmp = 0.0;
	if (y <= 1.95e-162)
		tmp = (y / x) / t_1;
	elseif (y <= 4.4e-10)
		tmp = (y / ((x - -1.0) * t_0)) * x;
	elseif (y <= 1.5e+118)
		tmp = (y / ((y - -1.0) * t_0)) * x;
	else
		tmp = ((x / (y + x)) / t_1) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, 1.95e-162], N[(N[(y / x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 4.4e-10], N[(N[(y / N[(N[(x - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.5e+118], N[(N[(y / N[(N[(y - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.95e-162

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 1.95e-162 < y < 4.3999999999999998e-10

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \cdot x} \]

   6. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   if 4.3999999999999998e-10 < y < 1.5e118

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 58.2

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 58.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + y\right)} \cdot x} \]

   6. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{y}{\left(y - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   if 1.5e118 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.1%

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

Alternative 16: 90.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) - -1}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, 2 + 3 \cdot y, y \cdot \left(1 + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-162)
   (/ (/ y x) (- (+ y x) -1.0))
   (if (<= y 7.5e-9)
     (* (/ y (* (- x -1.0) (* (+ y x) (+ y x)))) x)
     (if (<= y 3.2e+152)
       (/ x (fma x (+ 2.0 (* 3.0 y)) (* y (+ 1.0 y))))
       (/ (/ x (+ y x)) (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-162) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else if (y <= 7.5e-9) {
		tmp = (y / ((x - -1.0) * ((y + x) * (y + x)))) * x;
	} else if (y <= 3.2e+152) {
		tmp = x / fma(x, (2.0 + (3.0 * y)), (y * (1.0 + y)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-162)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) - -1.0));
	elseif (y <= 7.5e-9)
		tmp = Float64(Float64(y / Float64(Float64(x - -1.0) * Float64(Float64(y + x) * Float64(y + x)))) * x);
	elseif (y <= 3.2e+152)
		tmp = Float64(x / fma(x, Float64(2.0 + Float64(3.0 * y)), Float64(y * Float64(1.0 + y))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.95e-162], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e-9], N[(N[(y / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(x / N[(x * N[(2.0 + N[(3.0 * y), $MachinePrecision]), $MachinePrecision] + N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.95e-162

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 1.95e-162 < y < 7.49999999999999933e-9

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 58.5

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + x\right)} \cdot x} \]

   6. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   if 7.49999999999999933e-9 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \cdot \frac{x}{y + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \color{blue}{\frac{x}{y + x}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \cdot x \]

      12. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \frac{\left(y + x\right) - -1}{y}}} \]

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{y + 2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + \color{blue}{2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \color{blue}{\left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + \color{blue}{y}\right), y \cdot \left(1 + y\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

      6. lower-+.f64 52.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

   8. Applied rewrites 52.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)}} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 52.0

          \[\leadsto \frac{x}{\mathsf{fma}\left(x, 2 + 3 \cdot y, y \cdot \left(1 + y\right)\right)} \]

   11. Applied rewrites 52.0%

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

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) - -1}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, 2 + 3 \cdot y, y \cdot \left(1 + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-50)
   (/ (/ y x) (- (+ y x) -1.0))
   (if (<= y 3.2e+152)
     (/ x (fma x (+ 2.0 (* 3.0 y)) (* y (+ 1.0 y))))
     (/ (/ x (+ y x)) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else if (y <= 3.2e+152) {
		tmp = x / fma(x, (2.0 + (3.0 * y)), (y * (1.0 + y)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-50)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) - -1.0));
	elseif (y <= 3.2e+152)
		tmp = Float64(x / fma(x, Float64(2.0 + Float64(3.0 * y)), Float64(y * Float64(1.0 + y))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-50], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(x / N[(x * N[(2.0 + N[(3.0 * y), $MachinePrecision]), $MachinePrecision] + N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1000000000000002e-50

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 3.1000000000000002e-50 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \cdot \frac{x}{y + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \color{blue}{\frac{x}{y + x}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \cdot x \]

      12. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \frac{\left(y + x\right) - -1}{y}}} \]

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{y + 2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + \color{blue}{2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \color{blue}{\left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + \color{blue}{y}\right), y \cdot \left(1 + y\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

      6. lower-+.f64 52.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

   8. Applied rewrites 52.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)}} \]

   9. Taylor expanded in y around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 52.0

          \[\leadsto \frac{x}{\mathsf{fma}\left(x, 2 + 3 \cdot y, y \cdot \left(1 + y\right)\right)} \]

   11. Applied rewrites 52.0%

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

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) - -1}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, 3 \cdot y, y \cdot \left(1 + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-50)
   (/ (/ y x) (- (+ y x) -1.0))
   (if (<= y 3.2e+152)
     (/ x (fma x (* 3.0 y) (* y (+ 1.0 y))))
     (/ (/ x (+ y x)) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else if (y <= 3.2e+152) {
		tmp = x / fma(x, (3.0 * y), (y * (1.0 + y)));
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-50)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) - -1.0));
	elseif (y <= 3.2e+152)
		tmp = Float64(x / fma(x, Float64(3.0 * y), Float64(y * Float64(1.0 + y))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-50], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+152], N[(x / N[(x * N[(3.0 * y), $MachinePrecision] + N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1000000000000002e-50

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 3.1000000000000002e-50 < y < 3.20000000000000005e152

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \cdot \frac{x}{y + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \color{blue}{\frac{x}{y + x}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

      11. div-flip N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \cdot x \]

      12. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

   5. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \frac{\left(y + x\right) - -1}{y}}} \]

   6. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{y + 2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + \color{blue}{2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \color{blue}{\left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + \color{blue}{y}\right), y \cdot \left(1 + y\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

      6. lower-+.f64 52.0

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

   8. Applied rewrites 52.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)}} \]

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 51.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(x, 3 \cdot y, y \cdot \left(1 + y\right)\right)} \]

   11. Applied rewrites 51.6%

       \[\leadsto \frac{x}{\mathsf{fma}\left(x, 3 \cdot \color{blue}{y}, y \cdot \left(1 + y\right)\right)} \]

   if 3.20000000000000005e152 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-50) (/ (/ y x) (- (+ y x) -1.0)) (/ (/ x (+ y x)) (+ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 3.1d-50) then
        tmp = (y / x) / ((y + x) - (-1.0d0))
    else
        tmp = (x / (y + x)) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else {
		tmp = (x / (y + x)) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-50:
		tmp = (y / x) / ((y + x) - -1.0)
	else:
		tmp = (x / (y + x)) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-50)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) - -1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-50)
		tmp = (y / x) / ((y + x) - -1.0);
	else
		tmp = (x / (y + x)) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-50], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1000000000000002e-50

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 3.1000000000000002e-50 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      14. lower-*.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

      16. add-flip N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

      21. metadata-eval 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

      22. lift-+.f64 N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

      23. +-commutative N/A

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

      24. lower-+.f64 93.4

          \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   3. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot y}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      8. lower-/.f64 99.8

         \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \color{blue}{\frac{y}{y + x}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1} \cdot \frac{y}{y + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\left(y + x\right) - -1}} \cdot \frac{y}{y + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}} \cdot \frac{y}{y + x}}{\left(y + x\right) - -1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{y + x}}}{\left(y + x\right) - -1} \]

      6. frac-times N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(y + x\right) - -1} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) - -1} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{{\left(y + x\right)}^{2}}}}{\left(y + x\right) - -1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(y + x\right)}}^{2}}}{\left(y + x\right) - -1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{{\color{blue}{\left(x + y\right)}}^{2}}}{\left(y + x\right) - -1} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(y + x\right) - -1} \]

      14. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(y + x\right) - -1\right)} \]

      16. frac-times N/A

          \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(y + x\right) - -1}} \]

      17. div-flip N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{1}{\frac{\left(y + x\right) - -1}{y}}} \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{1}{\color{blue}{\frac{\left(y + x\right) - -1}{y}}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{\left(y + x\right) - -1}{y} \cdot \left(y + x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   9. Step-by-step derivation

      1. lower-+.f64 50.3

         \[\leadsto \frac{\frac{x}{y + x}}{1 + \color{blue}{y}} \]

   10. Applied rewrites 50.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 81.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-50) (/ (/ y x) (- (+ y x) -1.0)) (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 3.2d-50) then
        tmp = (y / x) / ((y + x) - (-1.0d0))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-50) {
		tmp = (y / x) / ((y + x) - -1.0);
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-50:
		tmp = (y / x) / ((y + x) - -1.0)
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-50)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) - -1.0));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-50)
		tmp = (y / x) / ((y + x) - -1.0);
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-50], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e-50

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 87.6

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      11. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      13. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      14. lower-+.f64 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot y}{\left(x + y\right) + 1} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)}} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      20. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)} \]

      21. metadata-eval 87.6

          \[\leadsto \frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - \color{blue}{-1}} \]

   3. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot y}{\left(y + x\right) - -1}} \]

   4. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]
   5. Step-by-step derivation

      1. lower-/.f64 51.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) - -1} \]

   6. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) - -1} \]

   if 3.2e-50 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.4

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      6. lower-/.f64 49.8

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + 1}}{y} \]

      9. add-flip N/A

         \[\leadsto \frac{\frac{x}{y - \left(\mathsf{neg}\left(1\right)\right)}}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

      11. lower--.f64 49.8

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

   6. Applied rewrites 49.8%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-50) (/ y (* x (+ 1.0 x))) (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-50) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    if (y <= 3.2d-50) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-50) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-50:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-50)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-50)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-50], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e-50

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 49.4

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.2e-50 < y

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.4

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      6. lower-/.f64 49.8

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + 1}}{y} \]

      9. add-flip N/A

         \[\leadsto \frac{\frac{x}{y - \left(\mathsf{neg}\left(1\right)\right)}}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

      11. lower--.f64 49.8

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

   6. Applied rewrites 49.8%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-149) (/ y (* x (+ 1.0 x))) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-149) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-149)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-149], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8000000000000001e-149

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(1 + x\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(1 + x\right)}} \]

      3. lower-+.f64 49.4

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -1.8000000000000001e-149 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. lower-+.f64 48.4

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{x}{y \cdot y + y} \]

      6. lower-fma.f64 48.4

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 48.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{\mathsf{fma}\left(y, y, y\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (fma y y y)))

C

assert(x < y);
double code(double x, double y) {
	return x / fma(y, y, y);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / fma(y, y, y))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   3. lower-+.f64 48.4

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 48.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{x}{y \cdot \left(y + \color{blue}{1}\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{1 \cdot y}} \]

   5. *-lft-identity N/A

      \[\leadsto \frac{x}{y \cdot y + y} \]

   6. lower-fma.f64 48.4

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
7. Add Preprocessing

Alternative 24: 3.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{2 \cdot x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* 2.0 x)))

C

assert(x < y);
double code(double x, double y) {
	return x / (2.0 * x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (2.0d0 * x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (2.0 * x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (2.0 * x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(2.0 * x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (2.0 * x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(2.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   10. +-commutative N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   14. lower-*.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot \left(x + y\right)} \]

   16. add-flip N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   17. lower--.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(x + y\right)} \]

   18. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(x + y\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   19. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   20. lower-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\color{blue}{\left(y + x\right)} - \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(x + y\right)} \]

   21. metadata-eval 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - \color{blue}{-1}\right) \cdot \left(x + y\right)} \]

   22. lift-+.f64 N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   23. +-commutative N/A

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

   24. lower-+.f64 93.4

       \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(y + x\right)}} \]

3. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)}} \cdot \frac{x}{y + x} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)} \cdot \color{blue}{\frac{x}{y + x}} \]

   5. frac-times N/A

      \[\leadsto \color{blue}{\frac{y \cdot x}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right) \cdot \left(y + x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(\left(y + x\right) - -1\right) \cdot \left(y + x\right)\right)} \cdot \left(y + x\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) - -1\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{y}{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x} \]

   11. div-flip N/A

       \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \cdot x \]

   12. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

   13. *-lft-identity N/A

       \[\leadsto \frac{\color{blue}{x}}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}} \]

   14. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(y + x\right) - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}{y}}} \]

5. Applied rewrites 84.7%

   \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \frac{\left(y + x\right) - -1}{y}}} \]

6. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{y + 2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + \color{blue}{2 \cdot \left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \color{blue}{\left(1 + y\right)}, y \cdot \left(1 + y\right)\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + \color{blue}{y}\right), y \cdot \left(1 + y\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

   6. lower-+.f64 52.0

      \[\leadsto \frac{x}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)} \]

8. Applied rewrites 52.0%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, y + 2 \cdot \left(1 + y\right), y \cdot \left(1 + y\right)\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{x}{2 \cdot \color{blue}{x}} \]
10. Step-by-step derivation

    1. lower-*.f64 3.5

       \[\leadsto \frac{x}{2 \cdot x} \]

11. Applied rewrites 3.5%

    \[\leadsto \frac{x}{2 \cdot \color{blue}{x}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025148 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64
  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))