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

Percentage Accurate: 69.6% → 99.8%

Time: 5.1s

Alternatives: 18

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. sqr-neg-rev N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-neg-in N/A

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

   13. sub-flip-reverse N/A

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

   14. lower--.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. frac-2neg N/A

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

   17. distribute-frac-neg2 N/A

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

   18. frac-2neg-rev N/A

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

   19. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{\frac{y}{y + x} \cdot x}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{y - \left(-1 - x\right)} \cdot x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5e-25)
   (/ (/ (* (/ y (+ y x)) x) (- x -1.0)) (+ y x))
   (/ (/ (* (/ y (- y (- -1.0 x))) x) (+ y x)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-25) {
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	} else {
		tmp = (((y / (y - (-1.0 - x))) * x) / (y + 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 <= 5d-25) then
        tmp = (((y / (y + x)) * x) / (x - (-1.0d0))) / (y + x)
    else
        tmp = (((y / (y - ((-1.0d0) - x))) * x) / (y + 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 <= 5e-25) {
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	} else {
		tmp = (((y / (y - (-1.0 - x))) * x) / (y + x)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5e-25:
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x)
	else:
		tmp = (((y / (y - (-1.0 - x))) * x) / (y + x)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5e-25)
		tmp = Float64(Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(Float64(Float64(y / Float64(y - Float64(-1.0 - x))) * x) / Float64(y + 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 <= 5e-25)
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	else
		tmp = (((y / (y - (-1.0 - x))) * x) / (y + 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, 5e-25], N[(N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999962e-25

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   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. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 75.5%

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

   if 4.99999999999999962e-25 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 89.0%

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

Alternative 3: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+130)
   (* (/ y (+ y x)) (/ x (* (- y (- -1.0 x)) (+ y x))))
   (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+130) {
		tmp = (y / (y + x)) * (x / ((y - (-1.0 - x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+130) then
        tmp = (y / (y + x)) * (x / ((y - ((-1.0d0) - x)) * (y + x)))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+130) {
		tmp = (y / (y + x)) * (x / ((y - (-1.0 - x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.7e+130:
		tmp = (y / (y + x)) * (x / ((y - (-1.0 - x)) * (y + x)))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+130)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y - Float64(-1.0 - x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+130)
		tmp = (y / (y + x)) * (x / ((y - (-1.0 - x)) * (y + x)));
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.7e+130], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e130

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   if 2.6999999999999998e130 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 4: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\frac{y}{y + x} \cdot x}{x - -1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-7)
   (/ (/ (* (/ y (+ y x)) x) (- x -1.0)) (+ y x))
   (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-7) {
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-7) then
        tmp = (((y / (y + x)) * x) / (x - (-1.0d0))) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-7) {
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-7:
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-7)
		tmp = Float64(Float64(Float64(Float64(y / Float64(y + x)) * x) / Float64(x - -1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-7)
		tmp = (((y / (y + x)) * x) / (x - -1.0)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-7], N[(N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-7

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   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. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 75.5%

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

   if 3.9999999999999998e-7 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 5: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{-1}{-1 - x} \cdot y}{x + y}\\ \mathbf{elif}\;x \leq -0.026:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y - -1\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+111)
   (/ (* (/ -1.0 (- -1.0 x)) y) (+ x y))
   (if (<= x -0.026)
     (* y (/ x (* (- x -1.0) (* (+ y x) (+ y x)))))
     (* (/ y (+ y x)) (/ x (* (- y -1.0) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+111) {
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	} else if (x <= -0.026) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (y / (y + x)) * (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 <= (-1.25d+111)) then
        tmp = (((-1.0d0) / ((-1.0d0) - x)) * y) / (x + y)
    else if (x <= (-0.026d0)) then
        tmp = y * (x / ((x - (-1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = (y / (y + x)) * (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 <= -1.25e+111) {
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	} else if (x <= -0.026) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.25e+111:
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y)
	elif x <= -0.026:
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))))
	else:
		tmp = (y / (y + x)) * (x / ((y - -1.0) * (y + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+111)
		tmp = Float64(Float64(Float64(-1.0 / Float64(-1.0 - x)) * y) / Float64(x + y));
	elseif (x <= -0.026)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y - -1.0) * Float64(y + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e+111)
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	elseif (x <= -0.026)
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	else
		tmp = (y / (y + x)) * (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, -1.25e+111], N[(N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.026], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2499999999999999e111

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   if -1.2499999999999999e111 < x < -0.0259999999999999988

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   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 75.8

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

      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 75.8

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

      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 75.8

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

      17. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right) \cdot \left(x + y\right)\right)} \]

      18. lower--.f64 N/A

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

      19. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      20. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right)\right)} \]

      21. lower--.f64 N/A

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

      22. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right)\right)} \]

   6. Applied rewrites 75.8%

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

   if -0.0259999999999999988 < x

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.3%

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

Alternative 6: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(x - -1\right) \cdot \left(y + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-7)
   (/ (* y (/ x (* (- x -1.0) (+ y x)))) (+ y x))
   (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-7) {
		tmp = (y * (x / ((x - -1.0) * (y + x)))) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-7) then
        tmp = (y * (x / ((x - (-1.0d0)) * (y + x)))) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-7) {
		tmp = (y * (x / ((x - -1.0) * (y + x)))) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-7:
		tmp = (y * (x / ((x - -1.0) * (y + x)))) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-7)
		tmp = Float64(Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(y + x)))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-7)
		tmp = (y * (x / ((x - -1.0) * (y + x)))) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-7], N[(N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-7

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 75.8%

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

   if 3.9999999999999998e-7 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 7: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-167}:\\ \;\;\;\;1 \cdot \frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;y \cdot \frac{x}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-206)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.56e-167)
     (* 1.0 (/ x (* (- y (- -1.0 x)) (+ y x))))
     (if (<= y 3.8e-7)
       (* y (/ x (* (- x -1.0) (* (+ y x) (+ y x)))))
       (/ (/ x (+ 1.0 y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-206) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.56e-167) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else if (y <= 3.8e-7) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-206) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.56d-167) then
        tmp = 1.0d0 * (x / ((y - ((-1.0d0) - x)) * (y + x)))
    else if (y <= 3.8d-7) then
        tmp = y * (x / ((x - (-1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-206) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.56e-167) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else if (y <= 3.8e-7) {
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7e-206:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.56e-167:
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)))
	elif y <= 3.8e-7:
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7e-206)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.56e-167)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(y - Float64(-1.0 - x)) * Float64(y + x))));
	elseif (y <= 3.8e-7)
		tmp = Float64(y * Float64(x / Float64(Float64(x - -1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-206)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.56e-167)
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	elseif (y <= 3.8e-7)
		tmp = y * (x / ((x - -1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7e-206], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.56e-167], N[(1.0 * N[(x / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-7], N[(y * N[(x / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.99999999999999979e-206

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 51.2

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

   11. Applied rewrites 51.2%

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

   if 6.99999999999999979e-206 < y < 1.56000000000000005e-167

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 93.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.0%

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

   if 1.56000000000000005e-167 < y < 3.80000000000000015e-7

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   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 75.8

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

      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 75.8

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

      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 75.8

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

      17. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right) \cdot \left(x + y\right)\right)} \]

      18. lower--.f64 N/A

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

      19. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right) \cdot \left(x + y\right)\right)} \]

      20. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(x + y\right)\right)\right)} \]

      21. lower--.f64 N/A

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

      22. lower--.f64 N/A

          \[\leadsto y \cdot \frac{x}{\left(x - \color{blue}{-1}\right) \cdot \left(\left(y + x\right) \cdot \mathsf{Rewrite<=}\left(lift-+.f64, \left(y + x\right)\right)\right)} \]

   6. Applied rewrites 75.8%

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

   if 3.80000000000000015e-7 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 8: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-167}:\\ \;\;\;\;1 \cdot \frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\left(x - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-206)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.56e-167)
     (* 1.0 (/ x (* (- y (- -1.0 x)) (+ y x))))
     (if (<= y 3.8e-7)
       (* (/ y (* (- x -1.0) (* (+ y x) (+ y x)))) x)
       (/ (/ x (+ 1.0 y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-206) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.56e-167) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else if (y <= 3.8e-7) {
		tmp = (y / ((x - -1.0) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-206) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.56d-167) then
        tmp = 1.0d0 * (x / ((y - ((-1.0d0) - x)) * (y + x)))
    else if (y <= 3.8d-7) then
        tmp = (y / ((x - (-1.0d0)) * ((y + x) * (y + x)))) * x
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-206) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.56e-167) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else if (y <= 3.8e-7) {
		tmp = (y / ((x - -1.0) * ((y + x) * (y + x)))) * x;
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7e-206:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.56e-167:
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)))
	elif y <= 3.8e-7:
		tmp = (y / ((x - -1.0) * ((y + x) * (y + x)))) * x
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7e-206)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.56e-167)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(y - Float64(-1.0 - x)) * Float64(y + x))));
	elseif (y <= 3.8e-7)
		tmp = Float64(Float64(y / Float64(Float64(x - -1.0) * Float64(Float64(y + x) * Float64(y + x)))) * x);
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-206)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.56e-167)
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	elseif (y <= 3.8e-7)
		tmp = (y / ((x - -1.0) * ((y + x) * (y + x)))) * x;
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7e-206], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.56e-167], N[(1.0 * N[(x / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-7], N[(N[(y / N[(N[(x - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.99999999999999979e-206

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 51.2

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

   11. Applied rewrites 51.2%

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

   if 6.99999999999999979e-206 < y < 1.56000000000000005e-167

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 93.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.0%

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

   if 1.56000000000000005e-167 < y < 3.80000000000000015e-7

   1. Initial program 69.6%

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

   2. Taylor expanded in y around 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 59.4

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

   4. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 72.0%

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

   if 3.80000000000000015e-7 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 9: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{-1}{-1 - x} \cdot y}{x + y}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{\left(y - -1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+25)
   (/ (* (/ -1.0 (- -1.0 x)) y) (+ x y))
   (if (<= x -1.8e-162)
     (* (/ x (* (- y -1.0) (* (+ y x) (+ y x)))) y)
     (/ (/ x (- y -1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+25) {
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	} else if (x <= -1.8e-162) {
		tmp = (x / ((y - -1.0) * ((y + x) * (y + x)))) * y;
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.5d+25)) then
        tmp = (((-1.0d0) / ((-1.0d0) - x)) * y) / (x + y)
    else if (x <= (-1.8d-162)) then
        tmp = (x / ((y - (-1.0d0)) * ((y + x) * (y + x)))) * y
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+25) {
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	} else if (x <= -1.8e-162) {
		tmp = (x / ((y - -1.0) * ((y + x) * (y + x)))) * y;
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.5e+25:
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y)
	elif x <= -1.8e-162:
		tmp = (x / ((y - -1.0) * ((y + x) * (y + x)))) * y
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+25)
		tmp = Float64(Float64(Float64(-1.0 / Float64(-1.0 - x)) * y) / Float64(x + y));
	elseif (x <= -1.8e-162)
		tmp = Float64(Float64(x / Float64(Float64(y - -1.0) * Float64(Float64(y + x) * Float64(y + x)))) * y);
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.5e+25)
		tmp = ((-1.0 / (-1.0 - x)) * y) / (x + y);
	elseif (x <= -1.8e-162)
		tmp = (x / ((y - -1.0) * ((y + x) * (y + x)))) * y;
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.5e+25], N[(N[(N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-162], N[(N[(x / N[(N[(y - -1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e25

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   if -2.50000000000000012e25 < x < -1.7999999999999999e-162

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 71.8%

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

   if -1.7999999999999999e-162 < x

   1. Initial program 69.6%

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

   2. Taylor expanded in 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 48.7

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

   4. Applied rewrites 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 50.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. add-flip-rev N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 50.0

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

   6. Applied rewrites 50.0%

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

Alternative 10: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+114}:\\ \;\;\;\;1 \cdot \frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-112)
   (/ (/ y (+ 1.0 x)) (+ x y))
   (if (<= y 1.1e+114)
     (* 1.0 (/ x (* (- y (- -1.0 x)) (+ y x))))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-112) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.1e+114) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-112) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else if (y <= 1.1d+114) then
        tmp = 1.0d0 * (x / ((y - ((-1.0d0) - x)) * (y + x)))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-112) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else if (y <= 1.1e+114) {
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.5e-112:
		tmp = (y / (1.0 + x)) / (x + y)
	elif y <= 1.1e+114:
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-112)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	elseif (y <= 1.1e+114)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(y - Float64(-1.0 - x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-112)
		tmp = (y / (1.0 + x)) / (x + y);
	elseif (y <= 1.1e+114)
		tmp = 1.0 * (x / ((y - (-1.0 - x)) * (y + x)));
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.5e-112], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+114], N[(1.0 * N[(x / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.50000000000000012e-112

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 51.2

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

   11. Applied rewrites 51.2%

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

   if 4.50000000000000012e-112 < y < 1.1e114

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 93.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.0%

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

   if 1.1e114 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 11: 81.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7.6e-57) (/ (/ y (+ 1.0 x)) (+ x y)) (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7.6e-57) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.6d-57) then
        tmp = (y / (1.0d0 + x)) / (x + y)
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7.6e-57) {
		tmp = (y / (1.0 + x)) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7.6e-57:
		tmp = (y / (1.0 + x)) / (x + y)
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7.6e-57)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.6e-57)
		tmp = (y / (1.0 + x)) / (x + y);
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7.6e-57], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5999999999999995e-57

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-out N/A

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

      12. metadata-eval N/A

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

      13. sub-flip N/A

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

      14. lift--.f64 N/A

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

      15. lower-/.f64 51.1

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

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 51.2

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

   11. Applied rewrites 51.2%

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

   if 7.5999999999999995e-57 < y

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.1

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

   6. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.1

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

      7. lift-/.f64 N/A

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

      8. frac-2neg N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}} \cdot y}{y + x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(1 + x\right)}\right)} \cdot y}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\left(1 + x\right)\right)} \cdot y}{y + x} \]

      11. distribute-neg-out N/A

          \[\leadsto \frac{\frac{-1}{\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}} \cdot y}{y + x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{-1}{-1 + \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)} \cdot y}{y + x} \]

      13. sub-flip N/A

          \[\leadsto \frac{\frac{-1}{-1 - \color{blue}{x}} \cdot y}{y + x} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{-1 - \color{blue}{x}} \cdot y}{y + x} \]

      15. lower-/.f64 51.1

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{y + x} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite<=}\left(+-commutative, \left(x + y\right)\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\frac{-1}{-1 - x} \cdot y}{x + y}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{x + y} \]

       2. lower-+.f64 50.6

          \[\leadsto \frac{\frac{x}{1 + \color{blue}{y}}}{x + y} \]

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 81.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e-77) (/ y (* x (+ 1.0 x))) (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-77) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d-77) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-77) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.6e-77:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-77)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-77)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.6e-77], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.59999999999999997e-77

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 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 48.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 4.59999999999999997e-77 < y

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}} \]

   3. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(y + x\right)}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{1 + x}} \]

      2. lower-+.f64 51.1

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{1 + \color{blue}{x}} \]

   6. Applied rewrites 51.1%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{1}{1 + x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{y + x}} \cdot \frac{1}{1 + x} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{1 + x}}{y + x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{1 + x}}{y + x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{1 + x} \cdot y}}{y + x} \]

      6. lower-*.f64 51.1

         \[\leadsto \frac{\color{blue}{\frac{1}{1 + x} \cdot y}}{y + x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{1 + x}} \cdot y}{y + x} \]

      8. frac-2neg N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}} \cdot y}{y + x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(1 + x\right)}\right)} \cdot y}{y + x} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{\mathsf{neg}\left(\left(1 + x\right)\right)} \cdot y}{y + x} \]

      11. distribute-neg-out N/A

          \[\leadsto \frac{\frac{-1}{\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}} \cdot y}{y + x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{-1}{-1 + \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)} \cdot y}{y + x} \]

      13. sub-flip N/A

          \[\leadsto \frac{\frac{-1}{-1 - \color{blue}{x}} \cdot y}{y + x} \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{-1 - \color{blue}{x}} \cdot y}{y + x} \]

      15. lower-/.f64 51.1

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{y + x} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(y + x\right)\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite<=}\left(+-commutative, \left(x + y\right)\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{-1 - x}} \cdot y}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(x + y\right)\right)} \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{\frac{-1}{-1 - x} \cdot y}{x + y}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{x + y} \]

       2. lower-+.f64 50.6

          \[\leadsto \frac{\frac{x}{1 + \color{blue}{y}}}{x + y} \]

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - -1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5e-56) (/ y (* x (+ 1.0 x))) (/ (/ x (- y -1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-56) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (y - (-1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (y - -1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5e-56:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (y - -1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5e-56)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(y - -1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-56)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (y - -1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5e-56], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999997e-56

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 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 48.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 4.99999999999999997e-56 < y

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in 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 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(1 + y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(1 + y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{x}{\left(1 + y\right) \cdot \color{blue}{y}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{\color{blue}{y}} \]

      6. lower-/.f64 50.0

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{1 + y}}{y} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{x}{y + 1}}{y} \]

      9. add-flip-rev N/A

         \[\leadsto \frac{\frac{x}{y - \left(\mathsf{neg}\left(1\right)\right)}}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

      11. lower--.f64 50.0

          \[\leadsto \frac{\frac{x}{y - -1}}{y} \]

   6. Applied rewrites 50.0%

      \[\leadsto \frac{\frac{x}{y - -1}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 78.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5e-56) (/ y (* x (+ 1.0 x))) (/ x (+ (* y y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-56) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = x / ((y * y) + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5e-56:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = x / ((y * y) + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5e-56)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / Float64(Float64(y * y) + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-56)
		tmp = y / (x * (1.0 + x));
	else
		tmp = x / ((y * y) + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5e-56], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999997e-56

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in y around 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 48.9

         \[\leadsto \frac{y}{x \cdot \left(1 + \color{blue}{x}\right)} \]

   4. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 4.99999999999999997e-56 < y

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in 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 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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-+.f64 N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      7. lower-*.f64 48.7

         \[\leadsto \frac{x}{y \cdot y + y} \]

   6. Applied rewrites 48.7%

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-144) (* (/ 1.0 x) y) (/ x (+ (* y y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-144) {
		tmp = (1.0 / x) * y;
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.4d-144) then
        tmp = (1.0d0 / x) * y
    else
        tmp = x / ((y * y) + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-144) {
		tmp = (1.0 / x) * y;
	} else {
		tmp = x / ((y * y) + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-144:
		tmp = (1.0 / x) * y
	else:
		tmp = x / ((y * y) + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-144)
		tmp = Float64(Float64(1.0 / x) * y);
	else
		tmp = Float64(x / Float64(Float64(y * y) + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-144)
		tmp = (1.0 / x) * y;
	else
		tmp = x / ((y * y) + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.4e-144], N[(N[(1.0 / x), $MachinePrecision] * y), $MachinePrecision], N[(x / N[(N[(y * y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000017e-144

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

   3. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot y} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x\right)}} \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(1 + x\right)}} \cdot y \]

      3. lower-+.f64 48.8

         \[\leadsto \frac{1}{x \cdot \left(1 + \color{blue}{x}\right)} \cdot y \]

   6. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]
   8. Step-by-step derivation

      1. lower-/.f64 25.9

         \[\leadsto \frac{1}{x} \cdot y \]

   9. Applied rewrites 25.9%

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]

   if 3.40000000000000017e-144 < y

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in 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 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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-+.f64 N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      7. lower-*.f64 48.7

         \[\leadsto \frac{x}{y \cdot y + y} \]

   6. Applied rewrites 48.7%

      \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{1}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-144) (* (/ 1.0 x) y) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-144) {
		tmp = (1.0 / x) * y;
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-144)
		tmp = Float64(Float64(1.0 / x) * y);
	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[y, 3.4e-144], N[(N[(1.0 / x), $MachinePrecision] * y), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.40000000000000017e-144

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

   3. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot y} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x\right)}} \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(1 + x\right)}} \cdot y \]

      3. lower-+.f64 48.8

         \[\leadsto \frac{1}{x \cdot \left(1 + \color{blue}{x}\right)} \cdot y \]

   6. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]
   8. Step-by-step derivation

      1. lower-/.f64 25.9

         \[\leadsto \frac{1}{x} \cdot y \]

   9. Applied rewrites 25.9%

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]

   if 3.40000000000000017e-144 < y

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in 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 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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 48.7

         \[\leadsto \frac{x}{\mathsf{fma}\left(y, \color{blue}{y}, y\right)} \]

   6. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e-185) (* (/ 1.0 x) y) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-185) {
		tmp = (1.0 / x) * y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.04d-185)) then
        tmp = (1.0d0 / x) * y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-185) {
		tmp = (1.0 / x) * y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.04e-185:
		tmp = (1.0 / x) * y
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-185)
		tmp = Float64(Float64(1.0 / x) * y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.04e-185)
		tmp = (1.0 / x) * y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.04e-185], N[(N[(1.0 / x), $MachinePrecision] * y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.03999999999999997e-185

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot x\right)} \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \cdot y} \]

   3. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\frac{x}{\left(y - \left(-1 - x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)} \cdot y} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(1 + x\right)}} \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(1 + x\right)}} \cdot y \]

      3. lower-+.f64 48.8

         \[\leadsto \frac{1}{x \cdot \left(1 + \color{blue}{x}\right)} \cdot y \]

   6. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \cdot y \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]
   8. Step-by-step derivation

      1. lower-/.f64 25.9

         \[\leadsto \frac{1}{x} \cdot y \]

   9. Applied rewrites 25.9%

      \[\leadsto \frac{1}{\color{blue}{x}} \cdot y \]

   if -1.03999999999999997e-185 < x

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. Taylor expanded in 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 48.7

         \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

   4. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{y}} \]
   6. Step-by-step derivation

      1. lower-/.f64 26.4

         \[\leadsto \frac{x}{y} \]

   7. Applied rewrites 26.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 26.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 69.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

2. Taylor expanded in 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 48.7

      \[\leadsto \frac{x}{y \cdot \left(1 + \color{blue}{y}\right)} \]

4. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{y}} \]
6. Step-by-step derivation

   1. lower-/.f64 26.4

      \[\leadsto \frac{x}{y} \]

7. Applied rewrites 26.4%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(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))))