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

Percentage Accurate: 69.6% → 99.8%

Time: 3.7s

Alternatives: 20

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: 69.6% 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

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

FPCore

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

C

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

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 = ((y * (x / (x + y))) / (x + y)) / ((y + x) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 74.2

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

3. Applied rewrites 74.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((y / (x + y)) * (x / (x + y))) / ((y + x) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 74.2

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

3. Applied rewrites 74.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (y * ((x / (x + y)) / (x + y))) / ((y + x) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   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. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 74.2

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

3. Applied rewrites 74.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-+.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 4: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.52 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot \frac{x}{x + y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.52e+150)
     (/ (/ y x) t_0)
     (if (<= x -2e-16)
       (/ (/ (* y x) (* (+ y x) (+ y x))) t_0)
       (/ (* (/ y (+ x y)) (/ x (+ x y))) (+ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.52e+150) {
		tmp = (y / x) / t_0;
	} else if (x <= -2e-16) {
		tmp = ((y * x) / ((y + x) * (y + x))) / t_0;
	} else {
		tmp = ((y / (x + y)) * (x / (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) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.52d+150)) then
        tmp = (y / x) / t_0
    else if (x <= (-2d-16)) then
        tmp = ((y * x) / ((y + x) * (y + x))) / t_0
    else
        tmp = ((y / (x + y)) * (x / (x + y))) / (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 (x <= -1.52e+150) {
		tmp = (y / x) / t_0;
	} else if (x <= -2e-16) {
		tmp = ((y * x) / ((y + x) * (y + x))) / t_0;
	} else {
		tmp = ((y / (x + y)) * (x / (x + y))) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.52e+150:
		tmp = (y / x) / t_0
	elif x <= -2e-16:
		tmp = ((y * x) / ((y + x) * (y + x))) / t_0
	else:
		tmp = ((y / (x + y)) * (x / (x + y))) / (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 (x <= -1.52e+150)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -2e-16)
		tmp = Float64(Float64(Float64(y * x) / Float64(Float64(y + x) * Float64(y + x))) / t_0);
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(x + y))) / 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 (x <= -1.52e+150)
		tmp = (y / x) / t_0;
	elseif (x <= -2e-16)
		tmp = ((y * x) / ((y + x) * (y + x))) / t_0;
	else
		tmp = ((y / (x + y)) * (x / (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_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.52e+150], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2e-16], N[(N[(N[(y * x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52e150

   1. Initial program 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 63.7

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

   3. Applied rewrites 63.7%

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

   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 92.5

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

   6. Applied rewrites 92.5%

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

   if -1.52e150 < x < -2e-16

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.9

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

   3. Applied rewrites 91.9%

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

   if -2e-16 < x

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 72.2

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

   3. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

Alternative 5: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) + 1}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot \frac{x}{x + y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+101)
   (/ (/ y x) (+ (+ y x) 1.0))
   (if (<= x -1.15e-10)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (/ (* (/ y (+ x y)) (/ x (+ x y))) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+101) {
		tmp = (y / x) / ((y + x) + 1.0);
	} else if (x <= -1.15e-10) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((y / (x + y)) * (x / (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 (x <= (-2d+101)) then
        tmp = (y / x) / ((y + x) + 1.0d0)
    else if (x <= (-1.15d-10)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((y / (x + y)) * (x / (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 (x <= -2e+101) {
		tmp = (y / x) / ((y + x) + 1.0);
	} else if (x <= -1.15e-10) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((y / (x + y)) * (x / (x + y))) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+101:
		tmp = (y / x) / ((y + x) + 1.0)
	elif x <= -1.15e-10:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((y / (x + y)) * (x / (x + y))) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+101)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) + 1.0));
	elseif (x <= -1.15e-10)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(x + y))) / 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 (x <= -2e+101)
		tmp = (y / x) / ((y + x) + 1.0);
	elseif (x <= -1.15e-10)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((y / (x + y)) * (x / (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[x, -2e+101], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-10], 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], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e101

   1. Initial program 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 68.1

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

   3. Applied rewrites 68.1%

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

   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 90.5

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

   6. Applied rewrites 90.5%

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

   if -2e101 < x < -1.15000000000000004e-10

   1. Initial program 89.6%

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

   if -1.15000000000000004e-10 < x

   1. Initial program 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 72.5

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

Alternative 6: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -1.78 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot t\_0}{\left(y + x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot t\_0}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -1.78e-9)
     (/ (* (/ y x) t_0) (+ (+ y x) 1.0))
     (/ (* (/ y (+ x y)) t_0) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -1.78e-9) {
		tmp = ((y / x) * t_0) / ((y + x) + 1.0);
	} else {
		tmp = ((y / (x + y)) * 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 / (x + y)
    if (x <= (-1.78d-9)) then
        tmp = ((y / x) * t_0) / ((y + x) + 1.0d0)
    else
        tmp = ((y / (x + y)) * 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 / (x + y);
	double tmp;
	if (x <= -1.78e-9) {
		tmp = ((y / x) * t_0) / ((y + x) + 1.0);
	} else {
		tmp = ((y / (x + y)) * t_0) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -1.78e-9:
		tmp = ((y / x) * t_0) / ((y + x) + 1.0)
	else:
		tmp = ((y / (x + y)) * t_0) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -1.78e-9)
		tmp = Float64(Float64(Float64(y / x) * t_0) / Float64(Float64(y + x) + 1.0));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * 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 / (x + y);
	tmp = 0.0;
	if (x <= -1.78e-9)
		tmp = ((y / x) * t_0) / ((y + x) + 1.0);
	else
		tmp = ((y / (x + y)) * 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[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.78e-9], N[(N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.78e-9

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 76.1

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

   3. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative 85.7

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

   8. Applied rewrites 85.7%

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

   if -1.78e-9 < x

   1. Initial program 69.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 72.5

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

   3. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 7: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.78 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot t\_0}{t\_1}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot t\_0}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ y x) 1.0)))
   (if (<= x -1.78e-9)
     (/ (* (/ y x) t_0) t_1)
     (if (<= x -3.8e-68)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
       (if (<= x -1.8e-226) (/ (* (/ y (+ x y)) t_0) 1.0) (/ (/ x y) t_1))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.78e-9) {
		tmp = ((y / x) * t_0) / t_1;
	} else if (x <= -3.8e-68) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.8e-226) {
		tmp = ((y / (x + y)) * t_0) / 1.0;
	} else {
		tmp = (x / y) / t_1;
	}
	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 = x / (x + y)
    t_1 = (y + x) + 1.0d0
    if (x <= (-1.78d-9)) then
        tmp = ((y / x) * t_0) / t_1
    else if (x <= (-3.8d-68)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (x <= (-1.8d-226)) then
        tmp = ((y / (x + y)) * t_0) / 1.0d0
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.78e-9) {
		tmp = ((y / x) * t_0) / t_1;
	} else if (x <= -3.8e-68) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.8e-226) {
		tmp = ((y / (x + y)) * t_0) / 1.0;
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = (y + x) + 1.0
	tmp = 0
	if x <= -1.78e-9:
		tmp = ((y / x) * t_0) / t_1
	elif x <= -3.8e-68:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif x <= -1.8e-226:
		tmp = ((y / (x + y)) * t_0) / 1.0
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.78e-9)
		tmp = Float64(Float64(Float64(y / x) * t_0) / t_1);
	elseif (x <= -3.8e-68)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (x <= -1.8e-226)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * t_0) / 1.0);
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.78e-9)
		tmp = ((y / x) * t_0) / t_1;
	elseif (x <= -3.8e-68)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (x <= -1.8e-226)
		tmp = ((y / (x + y)) * t_0) / 1.0;
	else
		tmp = (x / y) / t_1;
	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[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.78e-9], N[(N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, -3.8e-68], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-226], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.78e-9

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 76.1

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

   3. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative 85.7

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

   8. Applied rewrites 85.7%

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

   if -1.78e-9 < x < -3.80000000000000038e-68

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.1%

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

   if -3.80000000000000038e-68 < x < -1.79999999999999997e-226

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.6

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

   3. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{x + y}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.0%

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

   if -1.79999999999999997e-226 < x

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 69.6

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

   3. Applied rewrites 69.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 95.9

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

   6. Applied rewrites 95.9%

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

Alternative 8: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ t_1 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.78 \cdot 10^{-9}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{x}{x + y}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

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))) (t_1 (+ (+ y x) 1.0)))
   (if (<= x -1.78e-9)
     (/ t_0 t_1)
     (if (<= x -3.8e-68)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
       (if (<= x -1.8e-226) (/ (* t_0 (/ x (+ x y))) 1.0) (/ (/ x y) t_1))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.78e-9) {
		tmp = t_0 / t_1;
	} else if (x <= -3.8e-68) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.8e-226) {
		tmp = (t_0 * (x / (x + y))) / 1.0;
	} else {
		tmp = (x / y) / t_1;
	}
	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)
    t_1 = (y + x) + 1.0d0
    if (x <= (-1.78d-9)) then
        tmp = t_0 / t_1
    else if (x <= (-3.8d-68)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else if (x <= (-1.8d-226)) then
        tmp = (t_0 * (x / (x + y))) / 1.0d0
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.78e-9) {
		tmp = t_0 / t_1;
	} else if (x <= -3.8e-68) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else if (x <= -1.8e-226) {
		tmp = (t_0 * (x / (x + y))) / 1.0;
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + y)
	t_1 = (y + x) + 1.0
	tmp = 0
	if x <= -1.78e-9:
		tmp = t_0 / t_1
	elif x <= -3.8e-68:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	elif x <= -1.8e-226:
		tmp = (t_0 * (x / (x + y))) / 1.0
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	t_1 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.78e-9)
		tmp = Float64(t_0 / t_1);
	elseif (x <= -3.8e-68)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	elseif (x <= -1.8e-226)
		tmp = Float64(Float64(t_0 * Float64(x / Float64(x + y))) / 1.0);
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	t_1 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.78e-9)
		tmp = t_0 / t_1;
	elseif (x <= -3.8e-68)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	elseif (x <= -1.8e-226)
		tmp = (t_0 * (x / (x + y))) / 1.0;
	else
		tmp = (x / y) / t_1;
	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[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.78e-9], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[x, -3.8e-68], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-226], N[(N[(t$95$0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.78e-9

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 76.1

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

   3. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 85.7%

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

   if -1.78e-9 < x < -3.80000000000000038e-68

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.1%

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

   if -3.80000000000000038e-68 < x < -1.79999999999999997e-226

   1. Initial program 70.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.6

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

   3. Applied rewrites 70.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{x + y}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.0%

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

   if -1.79999999999999997e-226 < x

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 69.6

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

   3. Applied rewrites 69.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 95.9

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

   6. Applied rewrites 95.9%

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

Alternative 9: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 3.4 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_1}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x + y} \cdot t\_0}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ y x) 1.0)))
   (if (<= y 3.4e-243)
     (/ (/ y x) t_1)
     (if (<= y 2.2e-16) (/ (* (/ y (+ x y)) t_0) 1.0) (/ t_0 t_1)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.4e-243) {
		tmp = (y / x) / t_1;
	} else if (y <= 2.2e-16) {
		tmp = ((y / (x + y)) * t_0) / 1.0;
	} else {
		tmp = t_0 / t_1;
	}
	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 = x / (x + y)
    t_1 = (y + x) + 1.0d0
    if (y <= 3.4d-243) then
        tmp = (y / x) / t_1
    else if (y <= 2.2d-16) then
        tmp = ((y / (x + y)) * t_0) / 1.0d0
    else
        tmp = t_0 / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (y + x) + 1.0;
	double tmp;
	if (y <= 3.4e-243) {
		tmp = (y / x) / t_1;
	} else if (y <= 2.2e-16) {
		tmp = ((y / (x + y)) * t_0) / 1.0;
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = (y + x) + 1.0
	tmp = 0
	if y <= 3.4e-243:
		tmp = (y / x) / t_1
	elif y <= 2.2e-16:
		tmp = ((y / (x + y)) * t_0) / 1.0
	else:
		tmp = t_0 / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 3.4e-243)
		tmp = Float64(Float64(y / x) / t_1);
	elseif (y <= 2.2e-16)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) * t_0) / 1.0);
	else
		tmp = Float64(t_0 / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 3.4e-243)
		tmp = (y / x) / t_1;
	elseif (y <= 2.2e-16)
		tmp = ((y / (x + y)) * t_0) / 1.0;
	else
		tmp = t_0 / t_1;
	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[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 3.4e-243], N[(N[(y / x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 2.2e-16], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / 1.0), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.39999999999999996e-243

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 70.5

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

   3. Applied rewrites 70.5%

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

   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 96.4

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

   6. Applied rewrites 96.4%

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

   if 3.39999999999999996e-243 < y < 2.2e-16

   1. Initial program 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 75.3

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

   3. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{x + y}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 76.6%

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

   if 2.2e-16 < y

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 76.0

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

   3. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 84.1%

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

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -0.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

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) x)))
   (if (<= y -0.9)
     t_0
     (if (<= y 2.55e-175)
       (/ y (* (+ 1.0 x) x))
       (if (<= y 1.5e+17)
         (/ x (* (+ 1.0 y) y))
         (if (<= y 1.02e+63) t_0 (/ (/ x y) y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -0.9) {
		tmp = t_0;
	} else if (y <= 2.55e-175) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 1.5e+17) {
		tmp = x / ((1.0 + y) * y);
	} else if (y <= 1.02e+63) {
		tmp = t_0;
	} else {
		tmp = (x / y) / 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) / x
    if (y <= (-0.9d0)) then
        tmp = t_0
    else if (y <= 2.55d-175) then
        tmp = y / ((1.0d0 + x) * x)
    else if (y <= 1.5d+17) then
        tmp = x / ((1.0d0 + y) * y)
    else if (y <= 1.02d+63) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -0.9) {
		tmp = t_0;
	} else if (y <= 2.55e-175) {
		tmp = y / ((1.0 + x) * x);
	} else if (y <= 1.5e+17) {
		tmp = x / ((1.0 + y) * y);
	} else if (y <= 1.02e+63) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) / x
	tmp = 0
	if y <= -0.9:
		tmp = t_0
	elif y <= 2.55e-175:
		tmp = y / ((1.0 + x) * x)
	elif y <= 1.5e+17:
		tmp = x / ((1.0 + y) * y)
	elif y <= 1.02e+63:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) / x)
	tmp = 0.0
	if (y <= -0.9)
		tmp = t_0;
	elseif (y <= 2.55e-175)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	elseif (y <= 1.5e+17)
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	elseif (y <= 1.02e+63)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / x) / x;
	tmp = 0.0;
	if (y <= -0.9)
		tmp = t_0;
	elseif (y <= 2.55e-175)
		tmp = y / ((1.0 + x) * x);
	elseif (y <= 1.5e+17)
		tmp = x / ((1.0 + y) * y);
	elseif (y <= 1.02e+63)
		tmp = t_0;
	else
		tmp = (x / y) / 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] / x), $MachinePrecision]}, If[LessEqual[y, -0.9], t$95$0, If[LessEqual[y, 2.55e-175], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+17], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+63], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -0.900000000000000022 or 1.5e17 < y < 1.02e63

   1. Initial program 60.2%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 49.4

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

   4. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 62.2

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 62.2%

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

   if -0.900000000000000022 < y < 2.55000000000000027e-175

   1. Initial program 72.5%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 2.55000000000000027e-175 < y < 1.5e17

   1. Initial program 82.1%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 53.0

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

   4. Applied rewrites 53.0%

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

   if 1.02e63 < y

   1. Initial program 62.7%

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

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 82.6

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

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.8

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 87.8%

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

Alternative 11: 81.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

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 2.55e-175) (/ (/ y x) t_0) (/ (/ x (+ x y)) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2.55e-175) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / (x + y)) / t_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) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 2.55d-175) then
        tmp = (y / x) / t_0
    else
        tmp = (x / (x + y)) / t_0
    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 <= 2.55e-175) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 2.55e-175:
		tmp = (y / x) / t_0
	else:
		tmp = (x / (x + y)) / t_0
	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 <= 2.55e-175)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / Float64(x + y)) / t_0);
	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 <= 2.55e-175)
		tmp = (y / x) / t_0;
	else
		tmp = (x / (x + y)) / t_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] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2.55e-175], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.55000000000000027e-175

   1. Initial program 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 68.6

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

   3. Applied rewrites 68.6%

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

   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 92.4

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

   6. Applied rewrites 92.4%

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

   if 2.55000000000000027e-175 < y

   1. Initial program 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      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. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 77.4

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

   3. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{\left(y + x\right) + 1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + y}} \cdot \frac{x}{x + y}}{\left(y + x\right) + 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}} \cdot \frac{x}{x + y}}{\left(y + x\right) + 1} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x + y} \cdot \color{blue}{\frac{x}{x + y}}}{\left(y + x\right) + 1} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{\left(y + x\right) + 1} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{\left(y + x\right) + 1} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{x + y}}{\left(y + x\right) + 1} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\frac{x}{x + y}}}{x + y}}{\left(y + x\right) + 1} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{x + y}}}{x + y}}{\left(y + x\right) + 1} \]

      11. lift-+.f64 99.8

          \[\leadsto \frac{\frac{y \cdot \frac{x}{x + y}}{\color{blue}{x + y}}}{\left(y + x\right) + 1} \]

   7. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{\left(y + x\right) + 1} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{\left(y + x\right) + 1} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.8%

       \[\leadsto \frac{\frac{\color{blue}{x}}{x + y}}{\left(y + x\right) + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

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 2.55e-175) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 2.55e-175) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_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) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 2.55d-175) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    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 <= 2.55e-175) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 2.55e-175:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	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 <= 2.55e-175)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	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 <= 2.55e-175)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_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] + 1.0), $MachinePrecision]}, If[LessEqual[y, 2.55e-175], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.55000000000000027e-175

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 68.6

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   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 92.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 92.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) + 1} \]

   if 2.55000000000000027e-175 < y

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 77.4

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 74.5

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 74.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.55e-175) (/ (/ y x) (+ 1.0 x)) (/ (/ x y) (+ (+ y x) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-175) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = (x / y) / ((y + x) + 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) :: tmp
    if (y <= 2.55d-175) then
        tmp = (y / x) / (1.0d0 + x)
    else
        tmp = (x / y) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-175) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = (x / y) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.55e-175:
		tmp = (y / x) / (1.0 + x)
	else:
		tmp = (x / y) / ((y + x) + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.55e-175)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.55e-175)
		tmp = (y / x) / (1.0 + x);
	else
		tmp = (x / y) / ((y + x) + 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_] := If[LessEqual[y, 2.55e-175], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.55000000000000027e-175

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 68.6

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   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 92.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 92.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) + 1} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. lower-+.f64 92.3

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   9. Applied rewrites 92.3%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if 2.55000000000000027e-175 < y

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 77.4

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 74.5

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 74.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.55e-175) (/ (/ y x) (+ 1.0 x)) (/ (/ x y) (+ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-175) {
		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 <= 2.55d-175) 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 <= 2.55e-175) {
		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 <= 2.55e-175:
		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 <= 2.55e-175)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / y) / 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 <= 2.55e-175)
		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, 2.55e-175], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.55000000000000027e-175

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 68.6

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   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 92.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 92.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(y + x\right) + 1} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. lower-+.f64 92.3

         \[\leadsto \frac{\frac{y}{x}}{1 + \color{blue}{x}} \]

   9. Applied rewrites 92.3%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   if 2.55000000000000027e-175 < y

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 77.4

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 74.5

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 74.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]
   8. Step-by-step derivation

      1. lower-+.f64 74.2

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

   9. Applied rewrites 74.2%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -0.9:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{y}{\left(1 + x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -0.9)
   (/ (/ y x) x)
   (if (<= y 2.55e-175) (/ y (* (+ 1.0 x) x)) (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -0.9) {
		tmp = (y / x) / x;
	} else if (y <= 2.55e-175) {
		tmp = y / ((1.0 + x) * 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 <= (-0.9d0)) then
        tmp = (y / x) / x
    else if (y <= 2.55d-175) then
        tmp = y / ((1.0d0 + x) * 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 <= -0.9) {
		tmp = (y / x) / x;
	} else if (y <= 2.55e-175) {
		tmp = y / ((1.0 + x) * x);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -0.9:
		tmp = (y / x) / x
	elif y <= 2.55e-175:
		tmp = y / ((1.0 + x) * 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 <= -0.9)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 2.55e-175)
		tmp = Float64(y / Float64(Float64(1.0 + x) * x));
	else
		tmp = Float64(Float64(x / y) / 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 <= -0.9)
		tmp = (y / x) / x;
	elseif (y <= 2.55e-175)
		tmp = y / ((1.0 + x) * 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, -0.9], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.55e-175], N[(y / N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.900000000000000022

   1. Initial program 30.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 71.5

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 95.4

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 95.4%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -0.900000000000000022 < y < 2.55000000000000027e-175

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   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. *-commutative N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot \color{blue}{x}} \]

      4. lower-+.f64 91.7

         \[\leadsto \frac{y}{\left(1 + x\right) \cdot x} \]

   4. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 2.55000000000000027e-175 < y

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{\left(x + y\right) + 1} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      21. lower-+.f64 77.4

          \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   3. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]
   5. Step-by-step derivation

      1. lower-/.f64 74.5

         \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{\left(y + x\right) + 1} \]

   6. Applied rewrites 74.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\left(y + x\right) + 1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]
   8. Step-by-step derivation

      1. lower-+.f64 74.2

         \[\leadsto \frac{\frac{x}{y}}{1 + \color{blue}{y}} \]

   9. Applied rewrites 74.2%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{1 + y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -18000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -18000.0) (/ (/ y x) x) (/ x (* (+ 1.0 y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -18000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / ((1.0 + y) * 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 (x <= (-18000.0d0)) then
        tmp = (y / x) / x
    else
        tmp = x / ((1.0d0 + y) * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -18000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -18000.0:
		tmp = (y / x) / x
	else:
		tmp = x / ((1.0 + y) * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -18000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -18000.0)
		tmp = (y / x) / x;
	else
		tmp = x / ((1.0 + y) * 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[x, -18000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -18000

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 80.7

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 85.3

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 85.3%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -18000 < x

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   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. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. lower-+.f64 75.3

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot y} \]

   4. Applied rewrites 75.3%

      \[\leadsto \color{blue}{\frac{x}{\left(1 + y\right) \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 65.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e+63) (/ (/ y x) x) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+63) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / 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 <= 1.02d+63) then
        tmp = (y / x) / x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+63) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e+63:
		tmp = (y / x) / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e+63)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e+63)
		tmp = (y / x) / x;
	else
		tmp = (x / y) / 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, 1.02e+63], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e63

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 50.0

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

      5. lower-/.f64 52.5

         \[\leadsto \frac{\frac{y}{x}}{x} \]

   6. Applied rewrites 52.5%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if 1.02e63 < y

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 82.6

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 87.8

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 87.8%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.02e+63) (/ y (* x x)) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+63) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / 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 <= 1.02d+63) then
        tmp = y / (x * x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.02e+63) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.02e+63:
		tmp = y / (x * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.02e+63)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.02e+63)
		tmp = y / (x * x);
	else
		tmp = (x / y) / 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, 1.02e+63], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e63

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 50.0

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if 1.02e63 < y

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 82.6

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]

      5. lower-/.f64 87.8

         \[\leadsto \frac{\frac{x}{y}}{y} \]

   6. Applied rewrites 87.8%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -18000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -18000.0) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -18000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * 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 (x <= (-18000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -18000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -18000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -18000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -18000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * 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[x, -18000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -18000

   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 inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

      3. lower-*.f64 80.7

         \[\leadsto \frac{y}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -18000 < x

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

      3. lower-*.f64 53.8

         \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   4. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* y y)))

C

assert(x < y);
double code(double x, double y) {
	return x / (y * y);
}

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 * y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (y * y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (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), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   3. lower-*.f64 36.8

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

4. Applied rewrites 36.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(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))))