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

Percentage Accurate: 69.1% → 99.8%

Time: 6.1s

Alternatives: 17

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: 69.1% 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

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 69.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   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. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. div-add-rev N/A

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

   11. lift-+.f64 N/A

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

   12. lower-/.f64 99.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. sub-flip N/A

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

   17. lift--.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. associate-/l* N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. *-commutative N/A

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

   16. times-frac N/A

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

10. Applied rewrites 99.8%

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

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y - \left(-1 - x\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{\frac{x}{x - -1}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- y (- -1.0 x))))
   (if (<= y -1.7e+119)
     (* (/ y (+ x y)) (/ (/ x (- x -1.0)) (+ x y)))
     (if (<= y 2.9e+144)
       (/ (* (/ x (+ y x)) y) (* t_0 (+ y x)))
       (/ (/ x t_0) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y - (-1.0 - x);
	double tmp;
	if (y <= -1.7e+119) {
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	} else if (y <= 2.9e+144) {
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y - ((-1.0d0) - x)
    if (y <= (-1.7d+119)) then
        tmp = (y / (x + y)) * ((x / (x - (-1.0d0))) / (x + y))
    else if (y <= 2.9d+144) then
        tmp = ((x / (y + x)) * y) / (t_0 * (y + x))
    else
        tmp = (x / t_0) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y - (-1.0 - x);
	double tmp;
	if (y <= -1.7e+119) {
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	} else if (y <= 2.9e+144) {
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y - (-1.0 - x)
	tmp = 0
	if y <= -1.7e+119:
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y))
	elif y <= 2.9e+144:
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y - Float64(-1.0 - x))
	tmp = 0.0
	if (y <= -1.7e+119)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(Float64(x / Float64(x - -1.0)) / Float64(x + y)));
	elseif (y <= 2.9e+144)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y - (-1.0 - x);
	tmp = 0.0;
	if (y <= -1.7e+119)
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	elseif (y <= 2.9e+144)
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+119], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+144], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000007e119

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

   8. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 76.6

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

   10. Applied rewrites 76.6%

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

   if -1.70000000000000007e119 < y < 2.89999999999999998e144

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   if 2.89999999999999998e144 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 4: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.118:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{x + y}}{x - -1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.118)
   (/ (/ (* x (/ y (+ x y))) (- x -1.0)) (+ x y))
   (/ (/ x (- y (- -1.0 x))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.118) {
		tmp = ((x * (y / (x + y))) / (x - -1.0)) / (x + y);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.118d0) then
        tmp = ((x * (y / (x + y))) / (x - (-1.0d0))) / (x + y)
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.118) {
		tmp = ((x * (y / (x + y))) / (x - -1.0)) / (x + y);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.118:
		tmp = ((x * (y / (x + y))) / (x - -1.0)) / (x + y)
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.118)
		tmp = Float64(Float64(Float64(x * Float64(y / Float64(x + y))) / Float64(x - -1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.118)
		tmp = ((x * (y / (x + y))) / (x - -1.0)) / (x + y);
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 0.118], N[(N[(N[(x * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.11799999999999999

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 76.7%

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

   if 0.11799999999999999 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 5: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.118:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{\frac{x}{x - -1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.118)
   (* (/ y (+ x y)) (/ (/ x (- x -1.0)) (+ x y)))
   (/ (/ x (- y (- -1.0 x))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.118) {
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.118d0) then
        tmp = (y / (x + y)) * ((x / (x - (-1.0d0))) / (x + y))
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.118) {
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.118:
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y))
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.118)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(Float64(x / Float64(x - -1.0)) / Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.118)
		tmp = (y / (x + y)) * ((x / (x - -1.0)) / (x + y));
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 0.118], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.11799999999999999

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

   8. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 76.6

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

   10. Applied rewrites 76.6%

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

   if 0.11799999999999999 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 6: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 0.118:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{\left(x - -1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+119)
   (/ (/ y x) (+ y x))
   (if (<= y 0.118)
     (* y (/ (/ x (+ x y)) (* (- x -1.0) (+ x y))))
     (/ (/ x (- y (- -1.0 x))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+119) {
		tmp = (y / x) / (y + x);
	} else if (y <= 0.118) {
		tmp = y * ((x / (x + y)) / ((x - -1.0) * (x + y)));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.7d+119)) then
        tmp = (y / x) / (y + x)
    else if (y <= 0.118d0) then
        tmp = y * ((x / (x + y)) / ((x - (-1.0d0)) * (x + y)))
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+119) {
		tmp = (y / x) / (y + x);
	} else if (y <= 0.118) {
		tmp = y * ((x / (x + y)) / ((x - -1.0) * (x + y)));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.7e+119:
		tmp = (y / x) / (y + x)
	elif y <= 0.118:
		tmp = y * ((x / (x + y)) / ((x - -1.0) * (x + y)))
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+119)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (y <= 0.118)
		tmp = Float64(y * Float64(Float64(x / Float64(x + y)) / Float64(Float64(x - -1.0) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+119)
		tmp = (y / x) / (y + x);
	elseif (y <= 0.118)
		tmp = y * ((x / (x + y)) / ((x - -1.0) * (x + y)));
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.7e+119], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.118], N[(y * N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x - -1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000007e119

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 38.9

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

   6. Applied rewrites 38.9%

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

   if -1.70000000000000007e119 < y < 0.11799999999999999

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   6. Applied rewrites 76.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 83.8%

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

   if 0.11799999999999999 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 7: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(y - -1\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 0.000415:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.45e-260)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 8.2e-162)
     (/ (* (/ x (+ y x)) y) (* (- y -1.0) (+ y x)))
     (if (<= y 0.000415)
       (* y (/ x (* (- x -1.0) (* (+ x y) (+ x y)))))
       (/ (/ x (- y (- -1.0 x))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-260) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 8.2e-162) {
		tmp = ((x / (y + x)) * y) / ((y - -1.0) * (y + x));
	} else if (y <= 0.000415) {
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.45d-260) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 8.2d-162) then
        tmp = ((x / (y + x)) * y) / ((y - (-1.0d0)) * (y + x))
    else if (y <= 0.000415d0) then
        tmp = y * (x / ((x - (-1.0d0)) * ((x + y) * (x + y))))
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-260) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 8.2e-162) {
		tmp = ((x / (y + x)) * y) / ((y - -1.0) * (y + x));
	} else if (y <= 0.000415) {
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.45e-260:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 8.2e-162:
		tmp = ((x / (y + x)) * y) / ((y - -1.0) * (y + x))
	elif y <= 0.000415:
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))))
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-260)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 8.2e-162)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(y - -1.0) * Float64(y + x)));
	elseif (y <= 0.000415)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-260)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 8.2e-162)
		tmp = ((x / (y + x)) * y) / ((y - -1.0) * (y + x));
	elseif (y <= 0.000415)
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.45e-260], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-162], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(y - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000415], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.45e-260

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 1.45e-260 < y < 8.20000000000000039e-162

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 76.0%

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

   if 8.20000000000000039e-162 < y < 4.15e-4

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 76.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip-rev N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 76.1

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

   6. Applied rewrites 76.1%

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

   if 4.15e-4 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 8: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-259}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y \cdot \left(1 + \frac{x}{y}\right)}\\ \mathbf{elif}\;y \leq 0.000415:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.18e-259)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 4e-162)
     (/ (/ x (- y -1.0)) (* y (+ 1.0 (/ x y))))
     (if (<= y 0.000415)
       (* y (/ x (* (- x -1.0) (* (+ x y) (+ x y)))))
       (/ (/ x (- y (- -1.0 x))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.18e-259) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 4e-162) {
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	} else if (y <= 0.000415) {
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.18d-259) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 4d-162) then
        tmp = (x / (y - (-1.0d0))) / (y * (1.0d0 + (x / y)))
    else if (y <= 0.000415d0) then
        tmp = y * (x / ((x - (-1.0d0)) * ((x + y) * (x + y))))
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.18e-259) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 4e-162) {
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	} else if (y <= 0.000415) {
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.18e-259:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 4e-162:
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)))
	elif y <= 0.000415:
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))))
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.18e-259)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 4e-162)
		tmp = Float64(Float64(x / Float64(y - -1.0)) / Float64(y * Float64(1.0 + Float64(x / y))));
	elseif (y <= 0.000415)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.18e-259)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 4e-162)
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	elseif (y <= 0.000415)
		tmp = y * (x / ((x - -1.0) * ((x + y) * (x + y))));
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.18e-259], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-162], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000415], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.18e-259

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 1.18e-259 < y < 3.99999999999999982e-162

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 62.7

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

   9. Applied rewrites 62.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. metadata-eval N/A

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

       4. sub-flip N/A

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

       5. lift--.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if 3.99999999999999982e-162 < y < 4.15e-4

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 76.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 76.1

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip-rev N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 76.1

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

   6. Applied rewrites 76.1%

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

   if 4.15e-4 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 9: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-259}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y \cdot \left(1 + \frac{x}{y}\right)}\\ \mathbf{elif}\;y \leq 0.000415:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.18e-259)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 4e-162)
     (/ (/ x (- y -1.0)) (* y (+ 1.0 (/ x y))))
     (if (<= y 0.000415)
       (* (/ y (* (- x -1.0) (* (+ x y) (+ x y)))) x)
       (/ (/ x (- y (- -1.0 x))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.18e-259) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 4e-162) {
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	} else if (y <= 0.000415) {
		tmp = (y / ((x - -1.0) * ((x + y) * (x + y)))) * x;
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.18d-259) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 4d-162) then
        tmp = (x / (y - (-1.0d0))) / (y * (1.0d0 + (x / y)))
    else if (y <= 0.000415d0) then
        tmp = (y / ((x - (-1.0d0)) * ((x + y) * (x + y)))) * x
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.18e-259) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 4e-162) {
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	} else if (y <= 0.000415) {
		tmp = (y / ((x - -1.0) * ((x + y) * (x + y)))) * x;
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.18e-259:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 4e-162:
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)))
	elif y <= 0.000415:
		tmp = (y / ((x - -1.0) * ((x + y) * (x + y)))) * x
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.18e-259)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 4e-162)
		tmp = Float64(Float64(x / Float64(y - -1.0)) / Float64(y * Float64(1.0 + Float64(x / y))));
	elseif (y <= 0.000415)
		tmp = Float64(Float64(y / Float64(Float64(x - -1.0) * Float64(Float64(x + y) * Float64(x + y)))) * x);
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.18e-259)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 4e-162)
		tmp = (x / (y - -1.0)) / (y * (1.0 + (x / y)));
	elseif (y <= 0.000415)
		tmp = (y / ((x - -1.0) * ((x + y) * (x + y)))) * x;
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.18e-259], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-162], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000415], N[(N[(y / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.18e-259

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 1.18e-259 < y < 3.99999999999999982e-162

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 62.7

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

   9. Applied rewrites 62.7%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. metadata-eval N/A

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

       4. sub-flip N/A

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

       5. lift--.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if 3.99999999999999982e-162 < y < 4.15e-4

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 72.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip-rev N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 72.7

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

   6. Applied rewrites 72.7%

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

   if 4.15e-4 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 10: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-79)
   (/ (/ y (- x -1.0)) (+ y x))
   (if (<= y 7e-5)
     (/ (* x y) (* (* (+ x y) (+ x y)) 1.0))
     (/ (/ x (- y (- -1.0 x))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-79) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 7e-5) {
		tmp = (x * y) / (((x + y) * (x + y)) * 1.0);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d-79) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else if (y <= 7d-5) then
        tmp = (x * y) / (((x + y) * (x + y)) * 1.0d0)
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-79) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else if (y <= 7e-5) {
		tmp = (x * y) / (((x + y) * (x + y)) * 1.0);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.05e-79:
		tmp = (y / (x - -1.0)) / (y + x)
	elif y <= 7e-5:
		tmp = (x * y) / (((x + y) * (x + y)) * 1.0)
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-79)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	elseif (y <= 7e-5)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * 1.0));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-79)
		tmp = (y / (x - -1.0)) / (y + x);
	elseif (y <= 7e-5)
		tmp = (x * y) / (((x + y) * (x + y)) * 1.0);
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.05e-79], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-5], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-79

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if 1.05e-79 < y < 6.9999999999999994e-5

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 59.8

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.9%

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

   if 6.9999999999999994e-5 < y

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 11: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{y - -1}}{y + x}\\ \mathbf{if}\;x \leq -3.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (- y -1.0)) (+ y x))))
   (if (<= x -3.45e+62)
     (/ (/ y x) (+ y x))
     (if (<= x -5e+34) t_0 (if (<= x -1.8e-142) (/ y (* x (+ 1.0 x))) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x / (y - -1.0)) / (y + x);
	double tmp;
	if (x <= -3.45e+62) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e+34) {
		tmp = t_0;
	} else if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = 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 = (x / (y - (-1.0d0))) / (y + x)
    if (x <= (-3.45d+62)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-5d+34)) then
        tmp = t_0
    else if (x <= (-1.8d-142)) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x / (y - -1.0)) / (y + x);
	double tmp;
	if (x <= -3.45e+62) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e+34) {
		tmp = t_0;
	} else if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x / (y - -1.0)) / (y + x)
	tmp = 0
	if x <= -3.45e+62:
		tmp = (y / x) / (y + x)
	elif x <= -5e+34:
		tmp = t_0
	elif x <= -1.8e-142:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x / Float64(y - -1.0)) / Float64(y + x))
	tmp = 0.0
	if (x <= -3.45e+62)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5e+34)
		tmp = t_0;
	elseif (x <= -1.8e-142)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x / (y - -1.0)) / (y + x);
	tmp = 0.0;
	if (x <= -3.45e+62)
		tmp = (y / x) / (y + x);
	elseif (x <= -5e+34)
		tmp = t_0;
	elseif (x <= -1.8e-142)
		tmp = y / (x * (1.0 + x));
	else
		tmp = 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[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.45e+62], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+34], t$95$0, If[LessEqual[x, -1.8e-142], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.44999999999999999e62

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 38.9

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

   6. Applied rewrites 38.9%

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

   if -3.44999999999999999e62 < x < -4.9999999999999998e34 or -1.8e-142 < x

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.3

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

   6. Applied rewrites 50.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 50.3

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

   8. Applied rewrites 50.3%

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

   if -4.9999999999999998e34 < x < -1.8e-142

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 49.2

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

   4. Applied rewrites 49.2%

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

Alternative 12: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.45e+62)
   (/ (/ y x) (+ y x))
   (if (<= x -5e+34)
     (/ x (fma y y y))
     (if (<= x -1.8e-142) (/ y (* x (+ 1.0 x))) (/ (/ x y) (- y -1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.45e+62) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e+34) {
		tmp = x / fma(y, y, y);
	} else if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.45e+62)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5e+34)
		tmp = Float64(x / fma(y, y, y));
	elseif (x <= -1.8e-142)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.45e+62], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+34], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-142], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.44999999999999999e62

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 38.9

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

   6. Applied rewrites 38.9%

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

   if -3.44999999999999999e62 < x < -4.9999999999999998e34

   1. Initial program 69.1%

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

   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. lower-*.f64 N/A

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

      3. lower-+.f64 48.6

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

   4. Applied rewrites 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 48.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 48.6%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   if -4.9999999999999998e34 < x < -1.8e-142

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 49.2

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

   4. Applied rewrites 49.2%

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

   if -1.8e-142 < x

   1. Initial program 69.1%

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

   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. lower-*.f64 N/A

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

      3. lower-+.f64 48.6

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

   4. Applied rewrites 48.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 49.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip-rev N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 49.7

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

   6. Applied rewrites 49.7%

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

Alternative 13: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - \left(-1 - x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.4e-132)
   (/ (/ y (- x -1.0)) (+ y x))
   (/ (/ x (- y (- -1.0 x))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-132) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.4d-132)) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else
        tmp = (x / (y - ((-1.0d0) - x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-132) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.4e-132:
		tmp = (y / (x - -1.0)) / (y + x)
	else:
		tmp = (x / (y - (-1.0 - x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.4e-132)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y - Float64(-1.0 - x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.4e-132)
		tmp = (y / (x - -1.0)) / (y + x);
	else
		tmp = (x / (y - (-1.0 - x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.4e-132], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4000000000000004e-132

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -8.4000000000000004e-132 < x

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 51.1%

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

Alternative 14: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{y}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.4e-132)
   (/ (/ y (- x -1.0)) (+ y x))
   (/ (/ x (- y -1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-132) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y - -1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.4d-132)) then
        tmp = (y / (x - (-1.0d0))) / (y + x)
    else
        tmp = (x / (y - (-1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-132) {
		tmp = (y / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (y - -1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.4e-132:
		tmp = (y / (x - -1.0)) / (y + x)
	else:
		tmp = (x / (y - -1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.4e-132)
		tmp = Float64(Float64(y / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.4e-132)
		tmp = (y / (x - -1.0)) / (y + x);
	else
		tmp = (x / (y - -1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.4e-132], N[(N[(y / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4000000000000004e-132

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.5

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -8.4000000000000004e-132 < x

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.3

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

   6. Applied rewrites 50.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip-rev N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 50.3

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

   8. Applied rewrites 50.3%

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

Alternative 15: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-142) (/ y (* x (+ 1.0 x))) (/ (/ x y) (- y -1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y - -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 (x <= (-1.8d-142)) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / y) / (y - (-1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y - -1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-142:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (y - -1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-142)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - -1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-142)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / y) / (y - -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[x, -1.8e-142], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e-142

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 49.2

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

   4. Applied rewrites 49.2%

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

   if -1.8e-142 < x

   1. Initial program 69.1%

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

   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. lower-*.f64 N/A

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

      3. lower-+.f64 48.6

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

   4. Applied rewrites 48.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 49.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. add-flip-rev N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{x}{y}}{y - -1} \]

      10. lower--.f64 49.7

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

   6. Applied rewrites 49.7%

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

Alternative 16: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-142) (/ y (* x (+ 1.0 x))) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-142) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-142)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-142], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e-142

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 49.2

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

   4. Applied rewrites 49.2%

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

   if -1.8e-142 < x

   1. Initial program 69.1%

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

   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. lower-*.f64 N/A

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

      3. lower-+.f64 48.6

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

   4. Applied rewrites 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 48.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 48.6%

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return x / fma(y, y, y);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / fma(y, y, y))
end

Wolfram

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

Derivation

1. Initial program 69.1%

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

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. lower-*.f64 N/A

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

   3. lower-+.f64 48.6

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

4. Applied rewrites 48.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 48.6

      \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

6. Applied rewrites 48.6%

   \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(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))))